OFDM and SC-OFDM QLM

ABSTRACT

This invention provides a method for increasing the data rates supported by WiFi, WiMax, LTE communications using OFDM and SC-OFDM data symbol waveforms, by using quadrature layered modulation QLM which layers communications channels with a differentiating parameter for each layer that enables a demodulation algorithm to recover the data symbols in each layer, and supports higher data symbol rates then allowed by the Nyquist rate. A maximum likelihood (ML) QLM demodulation algorithm supports data rates to 4.75×57=271 Mbps compared to the current OFDM WiFi standard 57 Mbps with similar increases for WiMax, LTE. Multi-scale (MS) coding can be implemented to spread each data symbol over the OFDM band and over the 4 μs data packet to optimize BER performance. Computationally efficient signal processing for transmit and receive for OFDM and SC-OFDM are disclosed and Matlab direct error count Monte Carlo bit error rate simulations are evaluated to predict performance.

This patent filing is a continuation in part of application Ser. No.12/380,668 filed on Mar. 3, 2009.

BACKGROUND OF THE INVENTION

I. Field of the Invention

The present invention relates to cellular communications and alsorelates to the Nyquist rate for data symbol transmission, the Shannonbound on communications capacity, and symbol modulation and demodulationfor high-data-rate satellite, airborne, wired, wireless, and opticalcommunications and includes all of the communications symbol modulationsand the future modulations for single links and multiple access linkswhich include electrical and optical wired, mobile, point-to-point,point-to-multipoint, multipoint-to-multipoint, cellular, multiple-inputmultiple-output MIMO, and satellite communication networks. Inparticular it relates to WiFi, WiMax and long-term evolution LTE forcellular communications and satellite communications. WiFi, WiMax useorthogonal frequency division multiplexing OFDM on both links and LTEuses SC-OFDM on the uplink from user to base station and OFDM on thedownlink form base station to user. WiMax occupies a larger frequencyband than WiFi and both use OFDM waveforms. SC-OFDM is a single carrierorthogonal waveform version of OFDM which uses orthogonal frequencysubbands of varying widths.

II. Description of the Related Art

Two fundamental bounds on communications are the Nyquist rate and theShannon capacity theorem. The Nyquist rate is the complex digitalsampling rate equal to B that is sufficient to include all of theinformation within a frequency band B. For communications, equivalentexpressions for the Nyquist rate bound are defined in equations (1).T _(s)≧1/B  (1)BT_(S)≧1wherein 1/T_(s) is the data symbol transmission rate in the frequencyband B which means T_(s) is the spacing between the data symbols.

The Shannon bound for the maximum data rate C is a bound on thecorresponding number of information bits per symbol b as well as a boundon the communications efficiency η and is complemented by the Shannoncoding theorem, and are defined in equations (2).

$\begin{matrix}\begin{matrix}{{\text{Shannon~~bounds~~and~~coding~~theorem}\text{1~~Shannon~~capacity~~theorem}}\begin{matrix}{C = {{B\mspace{14mu}{\log_{2}\left( {1 + {S\text{/}N}} \right)}\mspace{14mu}\text{Channel~~capacity~~in~~bit/second}} = {Bps}}} \\{\text{for~~an~~additive~~~white~~Gaussian~~noise}\mspace{14mu}{AWGN}\text{~~channel~~with~~bandwidth}} \\{B\mspace{14mu}\text{wherein~~~}{``\log_{2}"}\mspace{14mu}\text{is~~the~~logarithm~~to~~the base~~2}} \\{= \text{Maximum~~~rate~~at~~which~~information~~can~~be reliably~~~transmitted~~~over}} \\{\text{a~~noisy~~channel~~where~~S/N~~is~~the signal-to-noise~~ratio~~in~~B}}\end{matrix}{{\text{2~~Shannon~~bound~~on~~b,}\mspace{14mu}\eta},\mspace{14mu}{\text{and}\mspace{14mu} E_{b}\text{/}N_{o}}}\begin{matrix}{{\max\left\{ b \right\}} = {\max\left\{ {C\text{/}B} \right\}}} \\{= {\log_{2}\left( {1 + {S\text{/}N}} \right)}} \\{= {\max\left\{ \eta \right\}}} \\{{E_{b}\text{/}N_{o}} = {{\left\lbrack {{{2\hat{}\max}\left\{ b \right\}} - 1} \right\rbrack/\max}\left\{ b \right\}}} \\{{\text{wherein~~}b} = {{C\text{/}B\mspace{14mu}{in}\mspace{14mu}{Bps}\text{/}{Hz}} = {{Bits}\text{/}\text{symbol}}}} \\{{\eta = {b\text{/}T_{s}B}},\mspace{14mu}{{Bps}\text{/}{Hz}}} \\{T_{s} = \text{symbol~~interval}}\end{matrix}} & (2)\end{matrix} & \;\end{matrix}$

3 Shannon coding theorem for the information bit rate R_(b)

-   -   For R_(b)<C there exists codes which support reliable        communications    -   For R_(b)>C there are no codes which support reliable        communications        wherein E_(b)/N_(o) is the ratio of energy per information bit        E_(b) to the noise power density N_(o), max{b} is the maximum        value of the number of information bits per symbol b and also is        the information rate in Bps/Hz, and since the communications        efficiency η=b/(T_(S)B) in bits/sec/Hz it follows that maximum        values of b and η are equal. Derivation of the equation for        E_(b)/N_(o) uses the definition E_(b)/N_(o)=(S/N)/b in addition        to 1 and 2. Reliable communications in the statement of the        Shannon coding theorem 3 means an arbitrarily low bit error rate        BER.

OFDM is defined in FIG. 1 for the WiFi 802.16 standard power spectrum in1,2 which implements the inverse FFT (IFFT=FFT⁻¹) to generate OFDM (orequivalently OFDMA which is orthogonal frequency division multipleaccess to emphasize the multiple access applications) data symbol tones2 over the first 3.2 μs of the 4 μS data packet in 30 in FIG. 7 withsome rolloff of the tones at their ends for spectral containment. Datasymbol tones are modulated with 4 PSK, 16QAM, 64QAM, 256QAM depending onthe transmission range and data rate and for 256QAM using the code rateoption R=¾ yields the information rate b=6 Bps/Hz for the WiFi standard,with other code options available. The N=64 point FFT⁻¹ generates N=64tones in 2 over the 20 MHz WiFi band with 48 tones used for datatransmission. In 3 the WiFi parameters are defined including acalculation of the maximum data rate R_(b)=57 Mbps. Later versions ofWiFi allow WiFi bands of 1.25, 5, 10, 20 MHz corresponding to N=4, 16,32, 64. For this representative OFDM WiFi QLM disclosure we areconsidering the WiFi standard in FIG. 1. The maximum data rate supportedby WiFi standard is calculated in 3 to be ˜57 Mbps using 256QAMmodulation and wherein “˜” represents an approximate value. OFDM usespulse waveforms in time and relies on the OFDM tone modulation toprovide orthogonality. SC-OFDM is a pulse-shaped OFDM that uses shapedwaveforms in time to roll-off the spectrum of the waveform betweenadjacent channels to provide orthogonality, allows the user to occupysubbands of differing widths, and uses a different tone spacing, datapacket length, and sub-frame length compared to OFDM for WiFi, WiMax.

SUMMARY OF THE INVENTION

This invention introduces a maximum likelihood ML demodulationarchitecture and implementation of a quadrature layered modulation QLMfor OFDM and SC-OFDM modulations to provide a method for increasing thedata rates. QLM for OFDM WiFi provides a method for increasing the datarates to 4.75× WiFi maximum data rate with current technology (4.75times the WiFi maximum rate) and to 6× WiFi maximum data rate withtechnology advances. QLM provides similar increases in data rate forOFDM WiMax and SC-OFDM LTE. QLM supports data symbol rates that can bemultiples of the Nyquist rate and communications data rates that can bemultiples of the Shannon bound.

A representative OFDM QLM architecture using ML demodulation isdisclosed in this invention for WiFi standard and the transmit andreceive signal processing algorithms and supporting block diagrams aredeveloped to illustrate the architecture and implementation. Thisarchitecture is directly applicable to WiMax by increasing the number ofOFDM tones to occupy the increased WiMax frequency band and also isdirectly applicable to SC-OFDM since the OFDM QLM ML architecture uses apulse-shaped OFDM which partitions the frequency band into orthogonalsubbands that can be combined to enable users to use differing frequencybands to implement SC-OFDM.

QLM is a layered topology for transmitting higher data rates thanpossible with a single layer of communications and is implemented bytransmitting each layer with a differentiating parameter which enablesseparation and decoding of each layer. For a representative OFDMarchitecture the OFDM WiFi QLM transmits the QLM signals over a set ofsubbands which together occupy the same frequency band as the WiFistandard 48 data symbol modulated FFT⁻¹ tones and over the WiFi 4 μsdata packet. Computationally efficient fast multi-channel FFT⁻¹ and FFTalgorithms generate the subband QLM data symbols for transmission andimplement the receive detection followed by maximum likelihood MLdemodulation, which can be implemented with a chip architecture thatsupports both OFDM WiFi and the OFDM WiFi QLM in this inventiondisclosure as well as both OFDM WiMax and OFDM WiMax QLM with anincrease in the frequency band, and with parameter changes the chiparchitecture supports both SC-OFDM LTE and SC-OFDM LTE QLM uplinks andboth OFDMA LTE and OFDMA LTE QLM downlinks. Monte Carlo Matlab directerror count bit-error-rate BER simulations for ML demodulation used inOFDM WiFi QLM demonstrate the QLM performance.

A multi-scale MS code can be implemented with modest complexity in orderto improve the bit-error-rate BER performance of OFDM WiFi QLM byspreading each transmitted data symbol over the OFDM WiFi QLM data bandand over the 4 μs data packet. Jensen's inequality from mathematicalstatistics proves that this uniform spreading of the Tx signals using MSprovides the best communications BER performance.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The above-mentioned and other features, objects, design algorithms, andperformance advantages of the present invention will become moreapparent from the detailed description set forth below when taken inconjunction with the drawings and performance data wherein likereference characters and numerals denote like elements and in which:

FIG. 1 describes the OFDM waveform for the WIFi standard.

FIG. 2 describes how to increase the data rate using a pulse waveform.

FIG. 3 describes how QLM increases the data rate for a pulse waveform ata constant frequency bandwidth.

FIG. 4 presents the equations for the communications capacity predictedby the new bound, the new Nyquist data symbol rate, and the BT_(S)product for a Wavelet waveform.

FIG. 5 calculates the ideal pulse correlation, candidate waveform Ψ timeresponse, and the Ψ correlation.

FIG. 6 presents the measured bit error rate BER performance of a pulsewaveform with 4 PSK modulation for QLM layers n_(p)=1,2,3,4 using aTrellis demodulation algorithm.

FIG. 7 describes the OFDM WiFi QLM 3-data symbol group and 4-data symbolgroup in each of the subbands and over the 4 μs WiFi data packet.

FIG. 8 presents the measured BER performance of a pulse waveform with 4PSK modulation for QLM layers n_(p)=1,2,4,8 for a 3-data symbol groupusing a maximum likelihood ML demodulation algorithm.

FIG. 9 presents the measured BER performance of a pulse waveform with 4PSK modulation for QLM layers n_(p)=1,2,4,8 for a 4-data symbol groupusing a ML demodulation algorithm.

FIG. 10 for the OFDM architecture calculates the information bits b persymbol interval equivalently expressed as Bps/Hz versus E_(b)/N_(o) forthe new bound, Shannon bound, for PSK, QAM with turbo coding to bringthe performance to essentially equal to the Shannon bound, and for QLM3,4-data symbol group performance using 64QAM.

FIG. 11 for the OFDM architecture calculates the information bits b persymbol interval expressed as Bps/Hz versus E_(b)/N_(o) for the newbound, Shannon bound, for PSK, QAM with turbo coding to bring theperformance to essentially equal to the Shannon bound, and for QLM3,4-data symbol group performance using 256QAM.

FIG. 12 for the OFDM architecture calculates the information bits b persymbol expressed as Bps/Hz versus S/N=C/I for the new bound, Shannonbound, for PSK, QAM with turbo coding to bring the performance toessentially equal to the Shannon bound, and for QLM 3,4-data symbolgroup performance using 64QAM.

FIG. 13 for the OFDM architecture calculates the information bits b persymbol interval expressed as Bps/Hz versus S/N=C/I for the new bound,Shannon bound, for PSK, QAM with turbo coding to bring the performanceto essentially equal to the Shannon bound, and for QLM 3,4-data symbolgroup performance using 256QAM.

FIG. 14 describes the partitioning of the OFDM WiFi standard 20 MHz bandinto 16 OFDM WiFi QLM subbands.

FIG. 15 presents the implementation steps in the transmitter to generatethe OFDM WiFi QLM baseband signal vector starting with the input datasymbols for the 4 μs data packet

FIG. 16 presents the implementation steps in the receiver to recover theestimated OFDM WiFi QLM input data symbols starting with the receivedOFDM WiFi QLM baseband signal vector for the 4 μs data packet.

FIG. 17 defines the complex Walsh codes and the generalized complexWalsh codes.

FIG. 18 describes the multi-scale MS encoding of each layer of OFDM WiFiQLM which spreads each data symbol vector within the subbands and overthe subbands.

FIG. 19 is a representative transmitter implementation block diagram forthe OFDM WiFi QLM mode.

FIG. 20 is a representative receiver implementation block diagram forthe OFDM WiFi QLM mode.

DETAILED DESCRIPTION OF THE INVENTION

OFDM and SC-OFDM applications of quadrature layered modulation QLM inthis invention disclosure are illustrated by the WiFi 802.16 standardwhich uses OFDM on both uplinks and downlinks between the user and basestation for cellular communications as well as for communications withsatellites. OFDM WiFi QLM replaces the OFDM orthogonal data symbol toneswith orthogonal subbands which are the same architecture as SC-OFDM usedfor the LTE uplink. This means the OFDM WiFi QLM architecture isdirectly applicable to WiMax by simply increasing the number of subbandssince they both use the same OFDM and 4 μs data packets with WiMax usinga larger frequency band, and also is directly applicable to the LTEuplink since the OFDM WiFi QLM orthogonal subbands partition thefrequency spectrum using the same architecture as SC-OFDM for LTE toallow the various users to be assigned differing orthogonal frequencysubbands across the frequency band and the QLM data symbol waveforms inthese subbands are SC-OFDM subband shaped waveforms used for LTE. TheOFDM WiFi QLM architecture is directly applicable to LTE downlinks whichuse OFDM.

OFDM WiFi QLM uses maximum likelihood ML demodulation of the quadraturelayered modulation QLM received correlated data symbols to support anarchitecture and implementation for QLM communications using the WiFi 4μs data packet over the 20 MHz WiFi band for the WiFi standard and withobvious extensions to the other WiFi versions.

FIG. 2 introduces QLM by considering an ideal pulse waveform in the timedomain. In 1 the pulse waveform is transmitted at the data symbol rateequal to 1/T_(s)=B where T_(s) is the pulse length, B is the bandwidth,the signal power level 2 is P=A² where “A” is the signal amplitude, andthe pulse modulation is phase shift keying PSK with “b” information bitsper data symbol. To increase the data symbol rate to n_(p)/T_(s) and theinformation rate to n_(p)b/T_(s), the pulse waveform is shortened 3 toT_(s)/n_(p) which increases the bandwidth to n_(p)B wherein B=1/T_(s)and requires the transmitted power to be increased 4 to P=n_(p)A² inorder to keep the same pulse energy per bit 5 is E_(b)=A²T_(s)/b. Thecorresponding energy-per-bit to noise power ratio 6 is E_(b)/N_(o)A²/2σ²b where 7 N_(o)=2σ²T_(s) is the noise power density and 2σ² is the“mean square” level of communication noise.

FIG. 3 implements this increase in the data symbol rate using QLMcommunications without changing the bandwidth of the pulse waveform byextending the pulses 8 in FIG. 2 over the original pulse length T_(s)and layering these extended data symbol waveforms on top of each other11 while occupying the same bandwidth B=1/T_(s). The pulse waveforms ineach layer 13 have E_(b)/N_(o) values equal to n_(p) times the originalE_(b)/N_(o)=A²/2σ²b due to the stretching of each pulse over T_(s)without changing the power level of the pulse. The layers are timesynchronized for transmission at ΔT_(s)=T_(s)/n_(p), 2ΔT_(s) . . . ,(n_(p)−1)ΔT_(s) offsets 14 respectively for layers 2, 3, . . . ,(n_(p)−1) relative to the 1^(st) layer at zero offset. This means thesignal-to-noise power S/N over B=1/T_(s) is equal to n_(p)^2 times theoriginal S/N due to the addition of the n_(p) pulse power levels 12 overeach T_(s) interval and the scaling of E_(b)/N_(o) by n_(p). Thisscaling of E_(b)/N_(o) in each of the layered communications channels issummarized in equation (3) along with the corresponding scaling of theS/N over T_(s). We find

$\begin{matrix}\begin{matrix}{{E_{b}/N_{o}} = {{n_{p}\left\lbrack {E_{b}/N_{o}} \right\rbrack}\mspace{14mu}{for}\mspace{14mu}{each}\mspace{14mu}{layer}\mspace{14mu}{or}\mspace{14mu}{channel}}} \\{= {n_{p}\left\lbrack {{A^{2}/2}\sigma^{2}b} \right\rbrack}} \\{{S/N} = {\sum{n_{p}{E_{b}/N_{o}}\mspace{14mu}{over}\mspace{14mu} n_{p}\mspace{14mu}{layer}\mspace{14mu}{or}\mspace{14mu}{channels}}}} \\{= {n_{p}\bigwedge{2\left\lbrack {S/N} \right\rbrack}}}\end{matrix} & (3)\end{matrix}$wherein [“o”] is the value of “o” for the communications channel whenthere is no layering.

FIG. 3 describes the layering of the communications channels for QLM andequation (3) defines the QLM scaling of the E_(b)/N_(o) and S/N. QLM isa layered topology for transmitting higher data rates than possible witheach layer of communications and is implemented by transmitting eachlayer with a differentiating parameter which enables separation anddecoding of each layer. Each layer or channel has a uniquedifferentiating parameter such as time offset as in FIG. 3 and/orfrequency offset. Each layer or channel obeys Shannon's laws when usingQLM scaling in equations (3).

The equations for the non-optimized channel capacity in Bps andinformation bits b per symbol interval are the Shannon's bounds in 1,2in equation (2) with the maximum “max” removed, with the [S/N] scalingin equations (3), and with the multiplication by “n_(p)” to account forthe n_(p) layers. We find

$\begin{matrix}\begin{matrix}{C = {n_{p}B\mspace{14mu}{\log_{2}\left\lbrack {1 + {\left( {S/N} \right)/{n_{p}\hat{}2}}} \right\rbrack}}} & {Bps} \\{b = {n_{p}\;{\log_{2}\left\lbrack {1 + {\left( {S/N} \right)/{n_{p}\hat{}2}}} \right\rbrack}}} & {{{Bps}/{Hz}} = {{{Bits}/{symbol}}\mspace{14mu}{interval}}}\end{matrix} & (4)\end{matrix}$using the definition b=CB in Bps/Hz=Bits/symbol from 2 in equations (2)and observing that “Bits/symbol” in 2 is “Bits/symbol interval” for QLMand wherein it is understood that the C,b are non-optimized values withrespect to the selection of the n_(p).

New upper bounds on C, b, η and a new lower bound on E_(b)/N_(o) arederived in equations (5) by using equation (4) and equation (2). We find

$\left. {\begin{matrix}{\text{New~~capacity~~bounds~~and coding theorem}{1\mspace{31mu} C} = {\max\left\{ {n_{p}B\mspace{14mu}{\log_{2}\left\lbrack {1 + {\left( {S\text{/}N} \right)\text{/}{n_{p}\hat{}2}}} \right\rbrack}} \right\}}} & (5)\end{matrix}\begin{matrix}{{2\mspace{40mu}\max\left\{ b \right\}}\; = {\max\left\{ {n_{p}{\log_{2}\left\lbrack {1 + {\left( {S\text{/}N} \right)\text{/}{n_{p}\hat{}2}}} \right\rbrack}} \right\}}} \\{= {\max\left\{ {n_{p}{\log_{2}\left\lbrack {1 + {\left( {b\mspace{14mu} E_{b}\text{/}N_{o}} \right){n_{p}\hat{}2}}} \right\rbrack}} \right\}}} \\{= {\max\left\{ \eta \right\}}}\end{matrix}{{3\mspace{34mu}\min\left\{ {E_{b}\text{/}N_{o}} \right\}} = {\min\left\{ \mspace{11mu}{{{\left\lbrack {{n_{p}\hat{}2}\text{/}b} \right\rbrack\left\lbrack {2\hat{}b} \right\}}\text{/}n_{p}} - 1} \right\rbrack}}} \right\}$4.  New  coding  theorem $\begin{matrix}{{\text{For}\mspace{14mu} R_{b}} < {C\mspace{11mu}\text{there exists codes which support reliable communications}}} \\{{\text{For}\mspace{11mu} R_{b}} > {C\mspace{14mu}\text{there are no codes which support reliable communications}}}\end{matrix}$ 5.  New  symbol  rate  n_(p)/T_(s) $\begin{matrix}{\mspace{104mu}{{\max\left\{ {n_{p}\text{/}T_{s}} \right\}} = {n_{p}B\mspace{14mu}{for}\mspace{14mu} n_{p}\mspace{14mu}{layers}\mspace{14mu}{of}\mspace{14mu}{communications}}}} \\{= {n_{p}x\mspace{14mu}\left( {{Nyquist}\mspace{14mu}{rate}\mspace{14mu}{for}\mspace{14mu} 1\mspace{14mu}{channel}} \right)}}\end{matrix}$wherein the maximum values of C, max{b}, and max{η} of C, b, η are therespective maximums of the expressions in equation (4) with respect ton_(p), the units of C, b, η are Bps, information bits/symbol interval,and Bps/Hz which means b is expressed in units Bps/Hz as well as inunits of information bits/symbol interval, and the min{E_(b)/N_(o)} isthe minimum of E_(b)/N_(o) with respect to n_(p) similar to thederivation in 2 in equations (2).

The new coding theorem in 4 in equations (5) states that C is the upperbound on the information data rate R_(b) in bits/second for which errorcorrecting codes exist to provide reliable communications with anarbitrarily low bit error rate BER wherein C is defined in 1 inequations (5) and upgrades the Shannon coding theorem 3 in equations (1)using new capacity bound C in 1 in equations (5) and introduces the newdata symbol rate 5 whose maximum value max{n_(p)/T_(s)} is n_(p) timesthe Nyquist rate for a single channel.

FIG. 4 restates the new communications bound in U.S. Pat. No. 7,391,819in a format suitable for implementations. Listed are the new Nyquistrate 25, new bounds on C, b, η in 26, and the assumed bandwidth-timeproduct in 27 with the note that the excess bandwidth a is zero α=0 fora QLM Wavelet waveform from U.S. Pat. No. 7,376,688 and filing Ser. No.12/152,318.

QLM demodulation received signal processing synchronizes and removes thereceived waveform by performing a convolution of the received waveformencoded data symbol with the complex conjugate of this waveform, todetect the correlated data symbols. This convolution is a correlation ofthe waveform with itself as illustrated in FIG. 5 since the waveformsare real and symmetric. These correlated data signals are processed witha trellis algorithm to recover estimates of the encoded symbol data, orprocessed by a ML algorithm to recover estimates of the data symbols, orprocessed by a recursive relaxation algorithm or another demodulationalgorithm to recover the transmitted data symbols.

FIG. 5 calculates the ideal triangular correlation 10, an examplewaveform 11 designated by Ψ, and the waveform Ψ correlation 12.Parameters of interest for this example square-root raised-cosinewaveform are the waveform length L=3, M=16, and excess bandwidth α=0.22,the mainlobe 13 which extends over a 2T_(s) interval, and the sidelobes14 which fall outside of the mainlobe. Parameter L is the waveformlength in units of M=16 samples and M is the number of digital samplesbetween adjacent waveforms at a Nyquist symbol rate=1/T_(s) for whichT_(s)=MT where 1/T is the digital sample rate and α is a measure of theroll-off of the frequency response. The ideal triangular correlation isthe correlation for the pulse waveform of length T_(s) in FIG. 3 andFIG. 5 demonstrates that for waveforms of interest for QLM thetriangular correlation approximates the mainlobe correlations for QLMwaveforms.

FIG. 6 measures the trellis decoding performance for uncoded 4-PSK forn_(p)=1 and for n_(p)=2,3,4 layers of QLM modulation by implementing thetrellis symbol decoding algorithm in U.S. Pat. No. 7,391,819 andvalidates the QLM technology and scaling. Performance is plotted as biterror rate BER versus the normalized value (E_(b)/N_(o))/n_(p) of theE_(b)/N_(o) for the new bound from equations (3),(5). Normalizationmeans that for a given BER the (E_(b)/N_(o))/n_(p) has the same valuefor all n_(p). For example, this means that BER=0.001 requires(E_(b)/N_(o))/n_(p)=6.8 dB and for n_(p)=1,2,4 this requiresE_(b)/N_(o)=6.8+0=6.8, 6.8+3=9.8, 6.8+6=12.8 dB respectively. Measuredperformance values for n_(p)=2,3,4 are from a direct error count MonteCarlo simulation of the trellis algorithm and are plotted in FIG. 6 asdiscrete measurement points.

FIG. 7 is a representative OFDM WiFi QLM architecture and implementationused to illustrate the implementation and performance. The OFDM WiFi QLMarchitecture partitions the WIFi 20 MHz frequency band into 16 1.25 MHzsubbands and uses the WiFi 4 μS data packet 30 to transmit the maximumlikelihood ML 3,4-data symbol groups in each OFDM WiFi QLM data subbandby partitioning the data packet into data processing blocks forimplementation of the OFDM WiFi QLM for transmission Tx using apost-weighted inverse fast fourier transform N-point FFT⁻¹ algorithm andfor the receiver Rx using a pre-weighted fast fourier transform N-pointFFT algorithm followed by a ML demodulation algorithm. These processingblocks 32 are the data symbol separation T_(s)=0.8 μs=NT for QLMtransmission in the OFDM WiFi QLM subbands wherein 1/T is the samplerate (bandwidth) of the FFT⁻¹ and FFT. Mainlobes of the representtivewaveforms in FIG. 5 for the 3,4-data symbol groups are depicted in 31and 33 and each extends over a 2Ts length as depicted in 13 in FIG. 5

In FIG. 7 the n_(s)=3-data symbol group 31 and the n_(s)=4-data symbolgroup 33 comprise 3 and 4 symbols labeled 1,2,3 and 1,2,3,4 respectivelyfor the QLM ground layer n_(p)=1. The data symbols for the additionallayers are overlayed as shown in FIG. 2 on data symbols 1,2 and 1,2,3and extend over data symbol 3 and 4 respectively. This layering meansthat for n_(p) layers the reference positions for the data symbols are1, 1+1/n_(p), 1+2/n_(p), . . . , 3 and 4 respectively corresponding todata symbols s=1, 2, . . . , N_(s)=(n_(s)−1)n_(p)+1 wherein n_(s) is thenumber of contiguous data symbols in the ground layer group, N_(s) isthe number of data symbols in the n_(s)-data symbol group for n_(p)layers, and the index s identifies the data symbols in the order theyare processed in the transmitter and in the receiver.

FIG. 7 OFDM WiFi QLM architecture for the n_(s)-data symbol groupsallows the ML system equations to be written in equation (6) for eachset of N_(s) detected correlated data symbols over the OFDM WiFi QLM 4μs data packet in each of the subbands k and enables a ML demodulationalgorithm to be implemented to recover the estimates of the datasymbols. System equations (6) calculate the received Rx detected datasymbol vector y(k) as a linear sum of the transmitted Tx data symbolvector x(k) multiplied by the correlation matrix H plus the Rx noisevector u(k). We findy(k)=H×(k)+u(k)  (6)

wherein

-   -   Y(k)=N_(s)×1 detected symbol vector. in subband k.    -   H=N_(s)×N_(s) correlation matrix of data symbols    -   x(k)=N_(s)×1 data symbol vector for layered channels in subband        k    -   u(k)=N_(s)×1 demodulation plus link noise vector for subband k        wherein the Rx data symbol vector y(k) elements are y(s|k) which        is the Rx detected correlated signal for state s=symbol s for        subband k, the data symbol vector x(k) elements are x(s|k) which        is the Tx data symbol x(s|k) for symbol s, u(k) is the noise        vector, and “x” is the multiply operator. In this disclosure the        same notation will be used for the estimates x(k), x(s|k) and        for the true values x(k),x(s|k) respectively with the        interpretation defined in the text. Equation (7) lists the ML        solution. We find

$\begin{matrix}\begin{matrix}{{1\mspace{14mu}{ML}\mspace{14mu}{Cost}\mspace{14mu}{{is}:\mspace{14mu} J}} = {\left\lbrack {{y(k)} - {{Hx}(k)}} \right\rbrack^{\prime}{Q^{- 1}\left\lbrack {{y(k)} - {{Hx}(k)}} \right\rbrack}}} \\{{= {( - ){exponent}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{ML}}}\mspace{25mu}} \\{{{probability}\mspace{20mu}{density}\mspace{14mu}{function}}\mspace{11mu}} \\{{2\mspace{14mu}{ML}\mspace{14mu}{solution}\mspace{14mu}{which}\mspace{14mu}{minimizes}\mspace{14mu} J\mspace{14mu}{{is}:{x(k)}}} = {\left\lbrack {H^{\prime}Q^{- 1}H} \right\rbrack^{- 1}H^{\prime}Q^{- 1}{y(k)}}} \\{{3\mspace{14mu}{The}\mspace{14mu}{noise}\mspace{14mu}{covariance}\mspace{14mu}{{is}:Q}} = {E\left\lbrack {{u(k)}{u(k)}^{\prime}} \right\}}} \\{{4\mspace{14mu}{ML}\mspace{14mu}{solution}\mspace{14mu}{simplifies}\mspace{14mu}{{to}:{x(k)}}} = {H^{- 1}{y(k)}}}\end{matrix} & (7)\end{matrix}$since the inverse H⁻¹ exists for all applications of interest, andwherein H′ is the conjugate transpose of H, and 2σ² is the mean-squaredata symbol detection noise.

FIGS. 8,9 measure the Monte Carlo bit error rate BER vs. the scaled(E_(b)/N_(o))/n_(p) from equations (3) for the 3-data symbol and 4-datasymbol groups in the OFDM WiFi QLM architecture in FIG. 7 withn_(p)=1,2,4,8 layers using 4 PSK symbol encoding and waveforms withideal correlations in FIG. 5 and the measured data validates the QLMtechnology and scaling. These BER measurements also apply to Wavelet andother waveforms since their correlations closely approximate idealcorrelations in FIG. 5. Measured BER performance losses compared toideal 4 PSK are expected to be the same for all data symbol modulationsincluding 8 PSK, 16QAM, 64QAM, 256QAM, 2048QAM since the ML demodulationestimates the data symbols in each layer independent of the data symbolmodulation. The 4 PSK modulation was used as a convenient modulation tomeasure the ML demodulation loss which loss is expected to apply to alldata symbol modulations.

FIGS. 10-13 calculate the OFDM WiFi QLM information rate b Bps/Hz(information bits per data symbol interval T_(s)) performance vs.E_(b)/N_(o) and S/N=C/I using the ML BER performance in FIGS. 8,9measured for the n_(s)=3,4-data symbol architectures in FIG. 7. FIG. 10calculates the b Bps/Hz vs. Eb/No for n_(s)=3,4-data symbol blocks usingQLM-64QAM which reads “modulation 64QAM used for QLM”, FIG. 11calculates the b Bps/Hz vs. Eb/No for n_(s)=3,4-data symbol blocks usingQLM-256QAM, FIG. 12 calculates the b Bps/Hz vs. S/N for n_(s)=3,4-datasymbol blocks using QLM-64QAM, and FIG. 13 calculates the b Bps/Hz vs.S/N for n_(s)=3,4-data symbol blocks using QLM-256QAM which reads“modulation 256QAM used for QLM”. Supporting this performance data arethe calculations for the Shannon bound in equations (2) and the newbound in equations (5), and the number of information bits per symbolb=bits=Bps/Hz versus measured E_(b)/N_(o), S/N=C/I for 4 PSK, 8 PSK,16QAM, 64QAM, 256QAM, 4096QAM. The 4 PSK, 8 PSK are 4-phase, 8-phasephase shift keying modulations which respectively encode 2,3 bits persymbol and 16QAM, 64QAM, 256QAM, 1024QAM are 16, 64, 256, 4096 state QAMmodulations which respectively encode 4, 6, 8, 12 bits. For no codingthe information bits per symbol b is equal to the modulation bits persymbol b_(s) so that b=b_(s)=2,3,4,6,8,12 bits per symbol=Bps/Hzrespectively for 4 PSK, 8 PSK, 16QAM, 64QAM, 256QAM, 4096QAM. Assumedturbo coding performance provides a performance almost equal to theShannon bound. The assumed coding rates R=3/4, 2/3, 3/4, 2/3, 3/4, 2/3reduce the information bits per symbol to the respective valuesb=1.5,2,3,4,6,8 bits=Bps/Hz.

The OFDM WiFi QLM performance calculations in FIGS. 10-13 use theinformation rates 1, scaling laws 2, and demodulation losses 3 inequations (8). OFDM WiFi QLM data rates 1 are calculated in FIGS. 10-13for the values of E_(b)/N_(o) and S/N calculated in 2 with demodulationlosses in 3 subtracted from these calculated values, forn_(p)=1,2,3,4,5,6,7,8 and QLM data symbol modulation choices.

$\begin{matrix}\begin{matrix}{{{1\mspace{14mu}{Information}\mspace{14mu}{rates}\mspace{14mu}{for}\mspace{14mu}{the}\mspace{14mu} n_{s}} = 3},{4 - {{data}\mspace{14mu}{symbol}\mspace{14mu}{groups}\mspace{14mu}{are}}}} \\{{b = {{b({QLM})}{N_{s}/4}\mspace{14mu}{for}\mspace{14mu} 3}},{4 -}} \\{{data}\mspace{14mu}{synbol}\mspace{14mu}{groups}\mspace{14mu}{wherein}} \\{N_{s} = {{{\left( {n_{s} - 1} \right)n_{p}} + 1} = {{number}\mspace{14mu}{of}\mspace{14mu}{symbols}}}} \\{{in}\mspace{14mu} 4\mspace{14mu}{\mu s}\mspace{14mu}{data}\mspace{14mu}{packet}} \\{{b({QLM})} = {{4\mspace{14mu}{{Bps}/{Hz}}\mspace{14mu}{for}\mspace{14mu}{QLM}} - {64{QAM}}}} \\{= {{6\mspace{14mu}{{Bps}/{Hz}}\mspace{14mu}{for}\mspace{14mu}{QLM}} - {256{QAM}}}} \\{{{divisor}\mspace{14mu} 4\mspace{14mu}{is}\mspace{14mu}{the}\mspace{14mu}{number}\mspace{14mu}{of}\mspace{14mu}{WiFi}}\mspace{14mu}} \\{{data}\mspace{14mu}{symbol}\mspace{14mu}{tones}\mspace{14mu}{being}\mspace{14mu}{replaced}} \\{{{by}\mspace{14mu}{the}\mspace{14mu}{QLM}\mspace{14mu} 3},{4 - {{data}\mspace{14mu}{symbol}}}} \\{groups}\end{matrix} & (8)\end{matrix}$

2 Scaling laws from equations (3) areE _(b) /N _(o) =[E _(b) /N _(o)]+10 log₁₀(n _(p))S/N=[S/N]+20 log₁₀(n _(p))

-   -   wherein [“o”] is the value of “o” for the communications channel        when there is no layering

3  E_(b)/N_(o)demodulation  loss  is $\begin{matrix}{{{E_{b}/N_{o}}\mspace{14mu}{loss}} = {{{- 1}\mspace{14mu}{db}\mspace{14mu}{for}\mspace{14mu} 3} - {{data}\mspace{14mu}{symbol}\mspace{14mu}{group}\mspace{14mu}{using}\mspace{14mu}{{FIG}.\mspace{14mu} 8}}}} \\{= {{{- 2}\mspace{14mu}{db}\mspace{14mu}{for}\mspace{14mu} 4} - {{data}\mspace{14mu}{symbol}\mspace{14mu}{group}\mspace{14mu}{using}\mspace{14mu}{{FIG}.\mspace{14mu} 9}}}}\end{matrix}$

OFDM WiFi QLM examples in equations (9) illustrate the performancecalculated in FIGS. 10-13 using equations (8). Information rate b inBps/Hz is calculated for n_(s)=3,4-data symbol blocks for n_(p)=6 layersfor QLM using the data symbol modulations 64QAM and 256QAM. A n_(p)=6layer QLM represents a reasonable implementation with current technologywhen considering the requirements on time and frequency synchronizationand ML demodulation. Also listed are the increase ΔS/N in signal powerthat is required to support this information rate when compared to theS/N=18 dB required to support the highest WiFi information rate b=6Bps/Hz using 256 QLM.

$\begin{matrix}\begin{matrix}{{{1\mspace{14mu}{OFDM}\mspace{14mu}{WiFi}\mspace{14mu}{QLM}\mspace{14mu}{for}\mspace{14mu} n_{s}} = {3 - {{data}\mspace{14mu}{symbol}\mspace{14mu}{block}}}},\;{n_{p} = {6\mspace{14mu}{layers}}}} \\{b = {{N_{s}/4}{x\left( {{WiFi}\mspace{14mu}{information}\mspace{14mu}{rate}} \right)}}} \\{{= {{3.25{x4}} = {13\mspace{14mu}{{Bps}/{Hz}}}}},{{{\Delta S}/N} = {8.8\mspace{14mu}{dB}}},} \\{{QLM} - {64{QAM}}} \\{{= {{3.25{x6}} = {19.5\mspace{14mu}{{Bps}/{Hz}}}}},{{{\Delta S}/N} =}} \\{{16.6\mspace{14mu}{dB}},{{QLM} - {256{QAM}}}} \\{{{2\mspace{14mu}{OFDM}\mspace{14mu}{WiFi}\mspace{14mu} Q\overset{\Cap}{L}M\mspace{14mu}{for}\mspace{14mu} n_{s}} = {4 - {{data}\mspace{14mu}{symbol}\mspace{14mu}{block}}}},\;{n_{p} = {6\mspace{14mu}{layers}}}} \\{b = {{N_{s}/4}{x\left( {{WiFi}\mspace{14mu}{information}\mspace{14mu}{rate}} \right)}}} \\{{= {{4.75{x4}} = {19\mspace{14mu}{{Bps}/{Hz}}}}},{{{\Delta S}/N} =}} \\{{9.8\mspace{14mu}{dB}},\;{{QLM} - {64{QAM}}}} \\{{= {{4.75{x6}} = {28.5\mspace{14mu}{{Bps}/{Hz}}}}},{{{\Delta S}/N} =}} \\{{17.6\mspace{14mu}{dB}},{{QLM} - {256{QAM}}}}\end{matrix} & (9)\end{matrix}$

The OFDM WiFi QLM maximum information rate b in equations (9) is listedin 1 in equations (10) and the corresponding Tx maximum data rate R_(b)vs the WiFi 256QAM maximum data rate is calculated in 2 in equations(10). We find

$\begin{matrix}{{{{QLM} - {256{QAM}\mspace{14mu}{data}\mspace{14mu}{rate}\mspace{14mu}{for}\mspace{14mu} 4} - {{data}\mspace{14mu}{symbol}\mspace{14mu}{group}\mspace{14mu}{with}\mspace{14mu} n_{p}}} = {6\mspace{14mu}{layers}}}\begin{matrix}{{\left. 1 \right)\mspace{14mu} b} = {28.5\mspace{14mu}{{Bps}/{Hz}}\mspace{14mu}{{vs}.{WiFi}}\mspace{14mu} 6\mspace{14mu}{{Bps}/{Hz}}\mspace{14mu}{maximum}\mspace{14mu}{information}\mspace{14mu}{rate}}} \\{{\left. 2 \right)R_{b}} = {4.75{x\left( {{WiFi}\mspace{14mu}{maximum}\mspace{14mu}{data}\mspace{14mu}{rate}\mspace{14mu} 57\mspace{14mu}{Mbps}} \right)}}} \\{= {271\mspace{14mu}{Mbps}\mspace{14mu}{{vs}.{WiFi}}\mspace{14mu}{maximum}\mspace{14mu}{data}\mspace{14mu}{rate}\mspace{14mu} 57\mspace{14mu}{Mbps}}}\end{matrix}} & (10)\end{matrix}$wherein the Tx maximum QLM-256QAM data rate R_(b) in Bps is scaled fromthe b for the WiFi maximum data rate calculated in 3 in FIG. 1.

FIG. 14 starts implementation of the OFDM WiFi QLM representativearchitecture with the generation of the subband filters by acomputationlly efficient N-point pre-weighted FFT⁻¹ algorithm. The 20MHz WiFi band 40 is partitioned into a 16 subband filters 41 which arelabeled form 1 to 16. Subbands 3-14 occupy the WiFi 48 data tonefrequency band and each subband filter occupies a 4-tone WiFi frequencyslot which means each subband filter occupies a frequency band=4/3.2μs=1.25 MHz and transmits a OFDM WiFi QLM data signal every T_(s)=3.2μs/4=0.8 μs when transmitting at the Nyquist rate 1/T_(s). The N can beequal to 16 or could be considerably larger if is necessary to generatemore than 16 digital samples per symbol length in order to generate anumber >16 which has a divisor equal to the n_(p) of interest so thatthe timing for each layer falls on a digital sample. Subbands 2 and 15each transmit the WiFi 2-tones at the edges of the data tone band. OFDMWiFi QLM architecture definitions include the following:

$\begin{matrix}\begin{matrix}{{x\left( s \middle| k \right)} = {{data}\mspace{14mu}{symbol}\mspace{14mu} s\mspace{14mu}{for}\mspace{14mu}{state}\mspace{14mu} s\mspace{14mu}{for}\mspace{14mu}{subband}\mspace{14mu} k}} \\{{n = 1},2,\ldots\mspace{14mu},{n_{p}\mspace{14mu}{index}\mspace{14mu}{over}\mspace{14mu}{the}\mspace{14mu} n_{p}\mspace{14mu}{layers}}} \\{n_{s} = {{{number}\mspace{14mu}{of}\mspace{14mu}{data}\mspace{14mu}{symbols}\mspace{14mu}{in}\mspace{14mu}{the}\mspace{14mu}{ground}\mspace{14mu}{layers}\mspace{14mu} n} = 1}} \\{{{Data}\mspace{14mu}{block}} = {{block} = {N - {{point}\mspace{14mu}{{FFT}^{- 1}/{FFT}}\mspace{14mu}{data}\mspace{14mu}{processing}\mspace{14mu}{block}}}}} \\{= {T_{s} = {{NT} = {0.8{\mu s}\mspace{14mu}{separation}\mspace{14mu}{between}\mspace{14mu}{symbols}\mspace{14mu}{within}}}}} \\{{each}\mspace{14mu}{layer}} \\{{1/T} = {{N/T_{s}} = {{{digital}\mspace{14mu}{sample}\mspace{14mu}{rate}} = {{digital}\mspace{14mu}{clock}\mspace{14mu}{rate}}}}} \\{{= {20\mspace{14mu}{and}\mspace{14mu} 30\mspace{14mu}{Mhz}\mspace{14mu}{options}\mspace{14mu}{for}\mspace{14mu}{FFT}^{- 1}}},{FFT}} \\{{s = 1},2,\ldots\mspace{14mu},{{N_{s}\mspace{14mu}{data}\mspace{14mu}{symbol}\mspace{14mu}{index}} = {{state}\mspace{14mu}{index}\mspace{14mu}{over}}}} \\{N_{s} = {{\left( {n_{s} - 1} \right)n_{p}} + 1}} \\{{k = 0},1,\ldots\mspace{14mu},{N - {1\mspace{14mu}{frequency}\mspace{14mu}{subband}\mspace{14mu}{index}\mspace{14mu}{in}\mspace{14mu}{the}\mspace{14mu} N} -}} \\{{point}\mspace{14mu}{FFT}^{- 1}\mspace{14mu}{and}\mspace{14mu}{FFT}} \\{{= 1},2,\ldots\mspace{14mu},{12\mspace{14mu}{subband}\mspace{14mu}{frequency}\mspace{14mu}{index}\mspace{14mu}{for}\mspace{14mu}{the}}} \\{{OFDM}\mspace{14mu}{WiFi}\mspace{14mu}{QLM}\mspace{14mu}{subbands}\mspace{14mu}{wherein}\mspace{14mu}{the}\mspace{14mu}{definition}} \\{{{of}\mspace{14mu}{the}\mspace{14mu}{frequency}\mspace{14mu}{index}\mspace{14mu} k\mspace{14mu}{is}\mspace{14mu}{determined}\mspace{14mu}{by}}\mspace{14mu}} \\{{the}\mspace{14mu}{application}} \\{\psi = {{data}\mspace{14mu}{symbol}\mspace{14mu}{x\left( s \middle| k \right)}\mspace{14mu}{waveform}\mspace{14mu}{represented}\mspace{14mu}{by}\mspace{14mu}{the}}} \\{{example}\mspace{14mu}{in}\mspace{14mu}{{FIG}.\mspace{14mu} 5}} \\{= {{real}\mspace{14mu}{symmetric}\mspace{14mu}{finite}\mspace{14mu}{impluse}\mspace{14mu}{response}\mspace{14mu}{in}\mspace{14mu}{time}\mspace{14mu}{which}}} \\{{is}\mspace{14mu} a\mspace{14mu}{Wavelet}\mspace{14mu}{waveform}\mspace{14mu}{or}\mspace{14mu}{equivalent}\mspace{14mu}{suitable}\mspace{14mu}{for}} \\{{the}\mspace{14mu}{FFT}^{- 1}{and}\mspace{14mu}{FFT}\mspace{14mu}{subbands}} \\{{i_{0} = 0},1,2,\ldots\mspace{14mu},{N - {1\mspace{14mu}{sample}\mspace{14mu}{index}\mspace{14mu}{over}\mspace{14mu}{data}\mspace{14mu}{block}\mspace{14mu}{for}}}} \\{{{the}\mspace{14mu} N} - {{point}\mspace{14mu}{FFT}^{- 1}{and}\mspace{14mu}{FFT}}} \\{{i = 0},1,2,\ldots\mspace{14mu},{{4\mspace{14mu}{\mu s}\mspace{14mu} T} - {1\mspace{14mu}{sample}\mspace{14mu}{index}\mspace{14mu}{and}\mspace{14mu}{clock}\mspace{14mu}{over}}}} \\{{the}\mspace{14mu} 4\mspace{14mu}{\mu s}\mspace{14mu}{data}\mspace{14mu}{packet}} \\{{= 80},{120\mspace{14mu}{digital}\mspace{14mu}{sample}\mspace{14mu}\left( {{digital}\mspace{14mu}{clocks}} \right)\mspace{14mu}{for}\mspace{14mu} 20},30} \\{{MHz} = {1/T}} \\{= {{i_{r} + {\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} = {i_{0}\mspace{14mu}{for}\mspace{14mu}{indexing}\mspace{14mu}{over}\mspace{14mu}{data}}}} \\{{symbol}\mspace{14mu} s\mspace{14mu}{waveform}\mspace{14mu}\psi} \\{{i_{r} = {{start}\mspace{14mu}{index}\mspace{14mu}{for}\mspace{14mu} 3}},{4 - {{data}\mspace{14mu}{symbol}\mspace{14mu}{group}\mspace{14mu}{waveform}}}} \\{{in}\mspace{14mu}{FIG}{.7}\mspace{14mu}{wherein}\mspace{14mu}{the}\mspace{14mu}{next}\mspace{14mu}{index}\mspace{14mu}{value}\mspace{14mu}{is}\mspace{14mu}{the}\mspace{14mu}{peak}} \\{{{{of}\mspace{14mu}{data}\mspace{14mu}{symbol}\mspace{11mu} s} = {1\mspace{14mu}{waveform}\mspace{14mu}\psi\mspace{14mu}{in}\mspace{14mu}{{FIG}.\mspace{14mu} 5}}}\mspace{14mu}} \\{{\left( {s - 1} \right){N/n_{p}}} = {{offset}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{start}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{data}\mspace{14mu}{symbol}\mspace{14mu} s\mspace{14mu}{waveform}}} \\{\psi\mspace{14mu}{referenced}\mspace{14mu}{to}\mspace{14mu}{the}\mspace{14mu}{peak}\mspace{14mu}{value}\mspace{14mu}{of}\mspace{14mu}\psi\mspace{14mu}{as}} \\{{{shown}\mspace{14mu}{in}\mspace{14mu}{{FIG}.\mspace{14mu} 5}},7} \\{{\Delta(s)} = {{data}\mspace{14mu}{block}\mspace{14mu}{offsets}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{data}\mspace{14mu}{symbol}\mspace{14mu} s\mspace{14mu}{waveform}}} \\{\psi\mspace{14mu}{relative}\mspace{14mu}{to}\mspace{14mu}{the}\mspace{14mu}{reference}\mspace{14mu}{block}\mspace{14mu}{corresponding}} \\{{{to}\mspace{14mu}{\Delta(s)}} = {0\mspace{14mu}{which}\mspace{14mu}{value}\mspace{14mu}{is}\mspace{14mu} 16\mspace{14mu}{in}\mspace{11mu}{{FIG}.\mspace{14mu} 5}\mspace{14mu}{and}}} \\{{block}\mspace{14mu} 2\mspace{14mu}{{FIG}.\mspace{14mu} 7}} \\{= {{offset}\mspace{14mu}{to}\mspace{14mu}{specify}\mspace{14mu}{the}\mspace{14mu}\psi\mspace{14mu}{waveform}\mspace{14mu}{indexed}\mspace{14mu}{by}\mspace{14mu} i_{0}}} \\{{for}\mspace{14mu}{processing}\mspace{14mu}{over}\mspace{14mu} a\mspace{14mu}{data}\mspace{14mu}{block}}\end{matrix} & (11)\end{matrix}$

Consider the generation of a data symbol x(s|k) waveform encodedbaseband signal z(i₀|s, Δ) for state s=signal s for processing block Δand for all of the subbands k. We find

-   -   z(i₀|s, Δ)=[1×N]_(s) baseband signal vector indexed on i₀ in        block Δ of (12) the waveform Ψencoded data symbols x(s|k) for        all k wherein w is a real symmetric Wavelet or equivalent        waveform impulse response suitable for post-weighted FFT⁻¹        subband data-filter waveforms

$\begin{matrix}\begin{matrix}{{\left. {{{z\left( i_{0} \right.}s},\Delta} \right) = {\left\lbrack {1{xN}} \right\rbrack_{s}\mspace{11mu}{baseband}\mspace{14mu}{signal}\mspace{14mu}{vector}\mspace{14mu}{indexed}\mspace{14mu}{on}\mspace{14mu} i_{0}\mspace{11mu}{in}{\mspace{11mu}\;}{block}}}{\mspace{14mu}\mspace{25mu}}} \\{\left. {\Delta\mspace{20mu}{of}\mspace{14mu}{the}\mspace{14mu}{waveform}\mspace{14mu}\psi\mspace{14mu}{encoded}\mspace{14mu}{data}\mspace{14mu}{symbols}\mspace{14mu}{x\left( s \right.}k} \right){\mspace{11mu}\;\mspace{20mu}}} \\{{{for}\mspace{20mu}{all}\mspace{14mu} k\mspace{20mu}{wherein}\mspace{14mu}\psi\mspace{14mu}{is}{\mspace{11mu}\;}a\mspace{14mu}{real}\mspace{14mu}{symmetric}\mspace{14mu}{Wavelet}\mspace{14mu}{or}}\mspace{31mu}} \\{{{equivalent}\mspace{14mu}{waveform}\mspace{11mu}{impulse}\mspace{14mu}{response}\mspace{14mu}{suitable}\mspace{14mu}{for}}{\mspace{14mu}\mspace{11mu}}} \\{{post}\text{-}{weighted}\mspace{14mu} F\; F\; T^{- 1}{subband}\mspace{14mu}{data}\text{-}{filter}\mspace{14mu}{waveforms}} \\{\left. {= {N^{- 1}\Sigma_{k}\mspace{11mu}{x\left( s \right.}k}} \right){\psi\left( {i_{0} + i_{r} + {\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N\left. k \right)}} \right.}} \\{{{using}\mspace{14mu}{the}\mspace{14mu}{definitions}{\mspace{11mu}\;}{in}\mspace{14mu}{equations}\mspace{14mu}(11)},(13)} \\{\left. {= {N^{- 1}\Sigma_{k}\mspace{11mu}{x\left( s \right.}k}} \right){E^{*}\left( {k,i_{0}} \right)}{\psi\left( {i_{0} + i_{r} + {\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} \right)}} \\{{{using}\mspace{14mu}{the}\mspace{14mu}{definition}\mspace{14mu}{of}\mspace{14mu}\psi\mspace{14mu}{in}\mspace{14mu}{{equations}{\mspace{11mu}\;}(11)}},(13)} \\{\left. {= {F\; F\; T^{- 1}\left\{ {{x\left( s \right.}k} \right)}} \right\}{\psi\left( {i_{0} + i_{r} + {\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} \right)}} \\{{using}\mspace{14mu}{the}\mspace{14mu}{definition}\mspace{14mu}{of}\mspace{14mu} F\; F\; T^{- 1}\mspace{14mu}{in}\mspace{14mu}{{equations}{\mspace{11mu}\;}(13)}}\end{matrix} & (12)\end{matrix}$which is a computationally efficient post-weighted FFT⁻¹{x(s|k)} thatcalculates a [1×N]_(x) row vector indexed on i₀ and with each element i₀multiplied by the corresponding element Ψ(i₀+i_(r)+(s−1)N/n_(p)+Δ(s)N)of the [1×N]_(Ψ) post-weighting Ψ row vector. It is convenient to usevector and matrix notation in order to map these algorithms intohardware chips. Definitions used are defined in equations (13).

$\quad\begin{matrix}\begin{matrix}{{{FFT}^{- 1}\left\{ {x\left( s \middle| k \right)} \right\}} = {N^{- 1}{\sum\limits_{k}{{x\left( s \middle| k \right)}\mspace{14mu} E*\left( {k,i_{0}} \right)\mspace{14mu}{by}\mspace{14mu}{definition}\mspace{14mu}{using}\mspace{14mu}{the}\mspace{14mu}{inverse}}}}} \\{{discrete}\mspace{14mu}{fourier}\mspace{14mu}{transform}\mspace{14mu}{DFT}^{- 1}\mspace{14mu}{equivalent}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{FFT}^{- 1}} \\{= {{\left\lbrack {1 \times 12} \right\rbrack\left\lbrack {12 \times N} \right\rbrack}\mspace{14mu}{multiplication}\mspace{14mu}{of}\mspace{14mu}{{the}\mspace{11mu}\left\lbrack {1 \times 12} \right\rbrack}\mspace{14mu}{row}\mspace{14mu}{vector}\mspace{14mu}{x\left( s \middle| k \right)}}} \\{{{for}\mspace{14mu}{the}\mspace{11mu} 12\mspace{14mu}{subbands}\mspace{14mu} k},\;{{by}\mspace{14mu}{{the}\;\left\lbrack {12 \times N} \right\rbrack}\mspace{14mu}{DFT}^{- 1}{matrix}}} \\{= {\left\lbrack {1 \times N} \right\rbrack_{x}\mspace{14mu}{row}\mspace{14mu}{vector}\mspace{14mu}{indexed}\mspace{14mu}{on}\mspace{14mu} i_{0}}}\end{matrix} & (13)\end{matrix}$

-   -   [1=N]_(Ψ)=post-weighting Ψ row vector indexed on i₀ with        elements        Ψ(i₀+i_(r)+(s−1)N/n_(p)+Δ(s)N)

$\begin{matrix}{{\psi\left( {i_{0} + i_{r} + {\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} \middle| k \right)} = {{waveform}\mspace{14mu}\psi\mspace{14mu}{for}\mspace{14mu}{subband}\mspace{14mu} k}} \\{= {{E*\left( {k,i_{0}} \right){\psi\left( {i_{0} + i_{r} + s - 1} \right)}{N/n_{p}}} +}} \\{\left. {{\Delta(s)}N} \right)\mspace{14mu}{since}\mspace{14mu}{the}\mspace{14mu}{baseband}\mspace{14mu}\psi} \\{{is}\mspace{14mu}{frequency}\mspace{14mu}{translated}\mspace{14mu}{to}} \\{{{subband}\mspace{14mu} k}\mspace{14mu}} \\{{E\left( {k,i_{0}} \right)} = {\exp\left( {{- {j2\pi}}\;{{ki}_{0}/N}} \right)}} \\{{= {{element}\mspace{14mu} k}},{i_{0}\mspace{14mu}{of}\mspace{14mu}{{the}\mspace{14mu}\left\lbrack {N \times N} \right\rbrack}}} \\{{DFT}\mspace{14mu}{matrix}\mspace{14mu}{and}\mspace{14mu} E*\left( {k,i_{0}} \right)\mspace{14mu}{is}} \\{{{the}\mspace{14mu}{complex}\mspace{14mu}{conjugate}\mspace{14mu}{of}}\mspace{14mu}} \\{{E\left( {k,i_{0}} \right)},{{harmonic}\mspace{14mu}{index}}} \\{{k = 0},1,\ldots\mspace{14mu},{N - 1},\mspace{14mu}{and}} \\{{time}\mspace{14mu}{index}\mspace{14mu} i_{0}}\end{matrix}$

FIG. 15 constructs the Tx baseband signal z(i) row vector [1×5N]_(z)wherein the number of indices of clocks in the 4 μdata packet is equalto N in each of the 5 blocks, using the implementation of thecomputationally efficient post-weighted FFT⁻¹ algorithm to generate thesubband QLM data symbols z(i₀|s; Δ) in equations (12),(13) for each s.Four implementation steps are used to generate the Tx [1×5N]_(z)baseband signal vector z(i). Step 1 in 11 starts the implementation bycalculating FFT⁻¹{x(s|k)}=[1×N]_(x) row vector indexed on i₀ inequations (13) for each s.

Step 2 in 12 calculates the Matlab element-by-element product “*” of the[1×N]_(x) row vector in step 1 with the [1×N}_(Ψ) row vector whoseelements are the Ψ values Ψ(i₀+i_(r)+(s−1)N/n_(p)+Δ(s)N) defined inequations (13), to implement the computationally efficient post-weightedFFT⁻¹ algorithm in equations (12) to calculate the [1×5N]_(s) row vectorz(i₀|s, Δ).

Step 3 in 13 unfolds the baseband waveform z(i₀|s, Δ) over all of thedata processing blocks Δ and adds zeros at both ends to expand thebaseband waveform to fill the OFDM WiFi QLM 4 μs data packet with theresultant [1×5N]_(s) baseband vector z(i|s) which is the Tx signal forstate s symbol s indexed by I over the 4 μs data packet. Vectorunfolding as described in 13 consists of applying the Matlab operationof forming the [1×5N]_(s) vector by first laying out the [1×N]_(s)vectors z(i₀|s, Δ) for all data processing blocks Δ and than adding zerorow vectors z(start) and z(finish) if necessary to complete the [1×5N] 4μs data packet vector z(i|s) for state s symbol s.

Step 4 in 14 is the final step in the implementation and generates theTx [1×5N]_(z) baseband signal vector z(i) for all of the states s andsignals s over the 4 μs OFDM WiFi QLM data packet by vector addition ofthe [1×5N]_(s) vectors z(i|s) over all values of state s symbol s.

The computationally complexity of the pre-weighted FFT⁻¹ algorithmincluding the addition of the vectors over the states and signals togenerate z(i) in Step 4 is the following wherein M=log₂(N). We find

$\begin{matrix}\begin{matrix}{{Real}\mspace{14mu}{multiplies}\mspace{14mu} R_{M}} & {R_{M} = {N_{s}\left( {{2{MN}} + {2{LN}}} \right)}} \\{{Real}{\;\mspace{11mu}}{adds}\mspace{14mu} R_{A}} & {R_{A} = {{N_{s}\left( {{3\mspace{11mu}{MN}} + {2{LN}}} \right)} + {\left( {N_{s} - 1} \right)2{LN}}}}\end{matrix} & (14)\end{matrix}$

The N-point post-weighted FFT⁻¹ for Tx and pre-weighted FFT for Rx havenominal values equal to N=16 which yields a 20 MHz digital sample (clockrate) equal to the WiFi 20 MHz band and N=16 digital samples over thedata symbol interval and which supports a layering menu n_(p)=1,2,4,8since these integers are divisors of N=16. To increase the menu ton_(r)=1,2,3,4,6,8 requires N=24 which yields a 30 MHz clock rate andN=24 digital samples over the data symbol interval and allows theintegers in the increased menu to be divisors of N=24. Higher values ofN are required to add n_(p)=5,7 values to the menu.

Rx demodulation of the received Tx signal z(i) row vector [1×5N] plusnoise starts by using pre-weighted FFT subband detection filters torecover the correlated data symbol estimates in the Tx signal plus noiseat the clock intervals N/n_(p) for the 3,4-data symbol groups layeredwith QLM communications channels in FIG. 7. This pre-weighted FFTsubband detection filter bank implements a convolution of the complexconjugate Ψ* of the Tx waveform Ψ in each of the subbands k at the clockintervals N/n_(p), with the received Tx signal z(i) to provide anoptimal detection which maximizes the S/N=C/I.

The correlated signals recovered by the pre-weighted FFT subbanddetection filters are characterized by the correlation coefficients{h(s, s′)} wherein s is the reference data symbol for state s which inour application are the data signals correlated with the transmitteddata signals s′ at the clock intervals N/n_(p). There areN_(s)=(n_(s)−1)n_(p)+1 QLM data symbols transmitted over each subband ofthe WiFi 4 μs packet and for both Tx and Rx signal processing it isconvenient to number them in the order they are received using thesignal index s for state index s in equations (11).

The correlation coefficients h(s,s′) are evaluated starting with thedefinition of the data symbols x(s|k), x(s′|k) waveforms using thedefinitions in equations (11),(13). We find

$\begin{matrix}\begin{matrix}{\left. {\left. {{\psi\left( {i_{0} + i_{r} + {\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} \right.}k} \right) = {{x\left( s \right.}k}} \right)\mspace{14mu}{waveform}} \\{\left. {\left. {{\psi\left( {i_{0} + i_{r} + {\left( {s^{\prime} - 1} \right){N/n_{p}}} + {{\Delta\left( s^{\prime} \right)}N}} \right.}k} \right) = {{x\left( s^{\prime} \right.}k}} \right)\mspace{14mu}{waveform}}\end{matrix} & (15)\end{matrix}$

The correlation coefficient h(s,s′) between data symbol x(s|k) and datasymbol x(s′|k) in the same subband k is by definition equal to

$\begin{matrix}\begin{matrix}{{h\left( {s,s^{\prime}} \right)} = {\sum\limits_{i\; 0}{\sum\limits_{\Delta{(s)}}{\psi*\left( {i_{0} + i_{r} + {\left( {s - 1} \right){N/n_{p}}} + {\Delta(s)N}} \middle| k \right){\psi\left( {i_{0} + i_{r} +} \right.}}}}} \\{\left. \left. {{\left( {s^{\prime} - 1} \right){N/n_{p}}} + {{\Delta\left( s^{\prime} \right)}N}} \middle| k \right. \right)\mspace{14mu}{wherein}\mspace{14mu}{the}\mspace{14mu}{convolution}\mspace{14mu}{has}} \\{{been}\mspace{14mu}{replaced}\mspace{14mu}{by}\mspace{14mu}{correlation}\mspace{14mu}{since}\mspace{14mu}{the}\mspace{14mu}\psi\mspace{14mu}{is}\mspace{14mu}{symmetric}} \\{= {\sum\limits_{i\; 0}{\sum\limits_{\Delta{(s)}}{\psi*\left( {i_{0} + i_{r} + {\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} \right){\psi\left( {i_{0} + i_{r} +} \right.}}}}} \\{{\left. {{\left( {s^{\prime} - 1} \right){N/n_{p}}} + {{\Delta\left( s^{\prime} \right)}N}} \right)\mspace{14mu}{since}\mspace{14mu} E*\left( {k,i_{0}} \right){E\left( {k,i_{0}} \right)}} = 1} \\{{and}\mspace{14mu}\psi\mspace{14mu}{is}\mspace{14mu}{real}\mspace{14mu}{and}\mspace{14mu}{symmetric}} \\{= {\sum\limits_{i}{{\psi(i)}{\psi\left( {i + {\left( {s - s^{\prime}} \right){N/n_{p}}} + {\left( {{\Delta(s)} - {\Delta\left( s^{\prime} \right)}} \right)N}} \right)}}}} \\{{which}\mspace{14mu}{is}\mspace{14mu}{an}\mspace{11mu}{equivalent}\mspace{14mu}{expression}\mspace{14mu}{using}\mspace{14mu}{index}\mspace{14mu} i\mspace{14mu}{over}\mspace{14mu} 4\mspace{14mu}\mu\; s} \\{{data}\mspace{14mu}{packet}}\end{matrix} & (16)\end{matrix}$wherein the correlation coefficients h(s,s′) are the row s and column s′elements of the [N_(s)xN_(s)]_(h) correlation matrix H=[h(s,s′)].

The computationally efficient pre-weighted FFT subband filters detecty(s|k) for state s in all of the subbands k by convolving the complexconjugate of the waveform of data symbol x(s|k) at state s in thereceiver with the Rx waveform z(i) in FIG. 7. We find

$\begin{matrix}\begin{matrix}{{y\left( s \middle| k \right)} = {\sum\limits_{i\; 0}{\sum\limits_{\Delta{(s)}}{{z\left( {i_{0} + i_{r} + {\left( {s^{\prime} - 1} \right){N/n_{p}}} + {{\Delta\left( s^{\prime} \right)}N}} \right)}\psi*\left( {i_{0} + i_{r} +} \right.}}}} \\\left. \left. {{\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} \middle| k \right. \right) \\{= {\sum\limits_{i\; 0}{\sum\limits_{\Delta{(s)}}{{z\left( {i_{0} + i_{r} + {\left( {s^{\prime} - 1} \right){N/n_{p}}} + {{\Delta\left( s^{\prime} \right)}N}} \right)}{\psi\left( {i_{0} + i_{r} +} \right.}}}}} \\{\left. {{\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} \right){E\left( {k,i_{0}} \right)}} \\{= {{FFT}{\underset{\Delta{(s)}}{\left\{ \left\lbrack \sum \right. \right.}{{z\left( {i_{0} + i_{r} + {\left( {s^{\prime} - 1} \right){N/n_{p}}} + {{\Delta\left( s^{\prime} \right)}N}} \right)}{\psi\left( {i_{0} + i_{r} +} \right.}}}}} \\\left. \left. \left. {{\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} \right) \right\rbrack \right\} \\{= {{FFT}\left\{ {{Pre} - {Sum}} \right\}}}\end{matrix} & (17)\end{matrix}$

wherein

$\begin{matrix}{{{Pre} - {Sum}} = {\underset{\Delta{(s)}}{\left\lbrack \sum \right.}{z\left( {i_{0} + i_{r} + {\left( {s^{\prime} - 1} \right){N/n_{p}}} + {{\Delta\left( s^{\prime} \right)}N}} \right)}{\psi\left( {i_{0} + i_{r} +} \right.}}} \\\left. \left. {{\left( {s - 1} \right){N/n_{p}}} + {{\Delta(s)}N}} \right) \right\rbrack \\{= {{\sum\limits_{s^{\prime}}{{h\left( {s,s^{\prime}} \right)}{x\left( s^{\prime} \middle| k \right)}}} + {{noise}\mspace{14mu}{using}\mspace{14mu}{equation}\mspace{14mu}(16)}}} \\{= {{x\left( s \middle| k \right)} + {\underset{s^{\prime}}{\left\lbrack \sum \right.}{h\left( {s,s^{\prime}} \right)}{x\left( s^{\prime} \middle| k \right)}_{{{for}\mspace{14mu} s^{\prime}} \neq \; s}} + {{noise}\mspace{14mu}{and}}}} \\{{{which}\mspace{14mu}{is}\mspace{14mu}{allowed}\mspace{14mu}{since}\mspace{14mu}{h\left( {s,s} \right)}} = {1\mspace{14mu}{because}\mspace{14mu}\psi\mspace{14mu}{is}}} \\{{{normalized}\mspace{14mu}{\sum\limits_{i}{{\psi(i)}{\psi(i)}}}} = 1}\end{matrix}$wherein the pre-sum is a [1×N]_(p) vector indexed on i₀ and the FFT ofthe pre-sum in equations (17) is a computationally efficientpre-weighted (pre-summed) FFT set of detection filters which calculatethe [1×12]_(y) vector whose elements are y(s|k) for k=1, 2, . . . , 12subbands by the matrix operation [1×N]_(p)[N×12]=[1×12]_(y) wherein[N×12] is the DFT matrix equivalent of the FFT.

FIG. 16 demodulates the Rx baseband signal z(i) row vector [1×5N]_(z)plus noise to recover the Rx estimates of the Tx data symbols x(s|k).Five implementation steps are used to recover the Rx estimates of x(s|k)starting with Step 1 in 21 which calculates the [1×N]_(p) pre-sumcomplex vector indexed on i₀ for state s symbol s in equations (17).

Step 2 in 22 calculates the FFT of this pre-sum vector to implement thecomputationally efficient pre-weighted FFT which generates the[1×12]_(y) correlated signal vector y(s)=[y(s|k=1) y(s|k=2) . . .y(s|k=12)] by the equivalent DFT [N×12] matrix multiplication[1×N]_(p)[N×12]=[1×12]₅, on the [1×1N]_(p) pre-sum vector in step 1.

Step 3 in 23 repeats steps 1 & 2 for all of the states s=1, 2, . . . ,N_(s)=(n_(s)−1) n_(p)+1 to generate the [N_(s)×12] correlated signalmatrix Y with [1×12]_(y) row vectors y(s) for each row s and [N_(x)×1]column vectors y(k)=[y(s=1|k); y(s=2|k); . . . ; y(N_(s)|k)} for eachcolumn k using Matlab operations to generate this column vector

Step 4 in 24 calculates the computationally efficient solution of the MLequation for each column y(k) of Y by solving x(k)=y(k) in 4 inequations (7) for the [N_(s)×1] column vector x(k)=[x(s=1|k); x(s=2|k);. . . ; x(s=N_(s)|k)] using Matlab operations to construct the columnvector whose elements x(s|k) are the Rx estimates for data symbol s forsubband k.

Step 5 in 25 recovers the data words for each estimate x(s|k) andsoft-decision decodes the data word bits to recover the information bitsin each data word.

The computational complexity of the pre-weighted FFT algorithm for the 4μs packet is the Following wherein M=log₂(N). We find

$\begin{matrix}\begin{matrix}{{Real}\mspace{14mu}{multiplies}\mspace{14mu} R_{M}} & {R_{M} = {N_{s}\left( {{2\mspace{14mu}{MN}} + {2{LN}}} \right)}} \\{{Real}{\;\mspace{11mu}}{adds}\mspace{14mu} R_{A}} & {R_{A} = {N_{s}\left( {{3\mspace{11mu}{MN}} + {2{LN}}} \right)}}\end{matrix} & (18)\end{matrix}$

It is well known that the ML solution has a fast algorithm whosecomputational complexity is estimated to be

$\begin{matrix}\begin{matrix}{{Real}\mspace{14mu}{multiplies}\mspace{14mu} R_{M}} & {R_{M} = {24\mspace{14mu} M_{s}N_{s}}} \\{{Real}{\;\mspace{11mu}}{adds}\mspace{14mu} R_{A}} & {R_{A} = {36\mspace{11mu} M_{s}N_{s}}}\end{matrix} & (19)\end{matrix}$wherein M_(s)=log₂ (N_(s)) and taking into account the number ofsubbands equal to 12. For a fast algorithm to apply it is necessary toremove one symbol from the n=1 ground layer in order to make the numberof symbols in a 4 μS packet equal to a product of primes rather than asingle prime.

Jensen's inequality from “Mathematical Statistics” by Fergeson, AcademicPress, 1967 is a fundamental lemma and when applied to OFDM WiFi QLMproves that a uniform spreading of the Tx signals using multi-scale MSencoding of each QLM layer of communicatons provides the bestcommunications BER performance. This means that there are no othercoding or spreading schemes for WiFi which can improve the BERperformance provided by MS OFDM WiFi QLM. MS encoding spreads each datasymbol over each 4 μs packet in each subband and simultaneously over allof the subbands. This MS OFDM WiFi QLM mode is disclosed using arepresentative implementation with complex Walsh CDMA codes andgeneralized complex Walsh CDMA codes and equally applies to allorthogonal and semi-orthogonal spreading codes.

FIG. 17 discloses a complex Walsh code and a generalized complex Walshcode which are used to implement the MS OFDM WiFi QLM mode. The complexWalsh will also refer to the generalized complex Walsh. An example ofthe complex Walsh code disclosed in U.S. Pat. No. 7,277,382, U.S. Pat.No. 7,352,796 is defined in FIG. 17 which defines 150 the complex N chipWalsh code vector W(c) indexed on c=0, 1, . . . , N−1 as a linear sum ofthe real code vector W(cr) lexicographically reordered by the index crand the real code vector W(ci) used as the complex orthogonal componentlexicographically reordered by the index ci and wherein the notation1:N/2 is the Matlab notation for indexing 1, 2, 3 . . . , N/2. Thisreordering of the even and odd Walsh vectors to yield the complex Walshcode vectors is in 1-to-1 correspondence with the generation of thecomplex discrete Fourier transform DFT codes by the reordered even andodd real DFT codes used for the real and imaginary complex DFT codecomponents. An 8×8 complex Walsh code matrix W₈ in 151 is rotated by 45degrees so the axes are aligned with the complex coordinates. Thegeneralized complex Walsh 154 is a tensor product with one or moreorthogonal matrices which include the discrete fourier transform (DFT),real Walsh, complex Walsh, and other codes. A motivation for thegeneralized Walsh is to obtain complex orthogonal codes with a greaterflexibility in choosing the length N.

FIG. 18 discloses an MS OFDM WiFi QLM code using complex Walsh codes andgeneralized complex Walsh codes and other codes. The MS coding isdisclosed in U.S. Pat. No. 7,394,792 and for QLM layer n=1, 2, . . . ,n_(p) in 101, MS maps the encoded OFDM WiFi QLM chips for layer n ontothe QLM frequency-time “f-t” communications space 105 by partitioning105 the WiFi band into N₁=12 subbands 104 using an orthogonal Waveletfilter bank or an equivalent filter bank and by mapping each data symbolover the N₁ subbands 104 and over the N₀ chips 103 in each subband. Forlayers n>1 the number of chips is N₀=n_(s)−1 and for the ground layern=1 the number of chips is N₀=n_(s) or N₀=n_(s)−1 if the last chip isdeleted in order to support a fast algorithm for ML demodulation. Powerspectral density PSD is plotted in 104 to illustrate the partitioninginto N₁=12 subbands. Parameters and coordinates in 102 are the numberN₀=N₀N₁ of MS chips, MS chip index n_(p) over the chips 103 in eachsubband, MS chip index n₁ over the subbands 104, and MS code indexn_(c). Index algebraic fields start with “0” and the layering index overn_(p) layers starts with the ground layer n=1. MS code and chip indexingin 102 illustrates the construction of an MS code index as the scaledalgebraic sum of the algebraic index fields for the user and chipindices over the communication elements being encoded. In this examplethe communication elements are the frequency band, individual subbands,and 4 μs data packet. This algebraic architecture spreads each user datasymbol uniformly over each member of each set of communication elementsthereby satisfying Jensen's inequality to guarantee the bestcommunications performance. The same construction is used for all of theQLM layers.

In the MS OFDM WiFi QLM mode the transmit data symbols x(s|n, ∀ k) ineach layer n are generated by encoding the transmit data symbols x(u|n)with the [N_(c)×N_(c)] MS code matrix C=[C(u₀+u₁N₀, n_(p)+n₁N₀)] priorto the transmit signal processing in FIG. 15 and in equations (12), (13)which implements a post-weighted FFT⁻¹ of these MS encoded data symbolsto generate the baseband OFDM WiFi QLM signal z(i).in step 4 in FIG. 15.Element C(u₀+u₁N₀, n_(p)+n₁N₀) is the element of C at row u₀+u₁N₀ andcolumn n_(p)+n₁N₀, column index n_(p)+n₁N₀ is the code index defined inFIG. 18, and user row index u₀+u₁N₀ has the same algebraic structure asthe code index. Symbol “∀k” reads “for all subbands k used by QLM datasymbols”. The set of transmit data symbols x(s|k,n) in layer n insubband k after MS encoding consists of the data symbols s=1, 2, . . . ,n_(s) for layer n=1 or s=1, 2, . . . , n_(s)−1 if the last symbol isdeleted to support a fast ML demodulation algorithm and the data symbolss=1, 2, . . . , n_(s)−1 for layer n>1. The transmit data symbolsx(s|n,k) in each layer n are generated by the MS encoding in equation(20) of the data symbols x(u|n) for u in layer n and which are prior tothe MS encoding.

$\begin{matrix}\begin{matrix}{{x\left( {{s = \left. {n_{0} + 1} \middle| n \right.},{k = {n_{1} + 1}}} \right)} = {\sum\limits_{u}{{x\left( u \middle| n \right)}{C\left( {{u_{0} + {u_{1}N_{0}}},{n_{0} + {n_{1}N_{0}}}} \right)}}}} \\{= {{MS}\mspace{14mu}{encoded}\mspace{14mu}{transmit}\mspace{14mu}{data}\mspace{14mu}{symbol}}} \\{s = {{u_{0} + {1\mspace{14mu}{in}\mspace{14mu}{subband}\mspace{14mu} k}} = {u_{1} + 1}}} \\{{transmit}\mspace{14mu}{MS}\mspace{14mu}{data}\mspace{14mu}{symbol}\mspace{14mu} s\mspace{14mu}{in}\mspace{14mu}{layer}} \\{n\mspace{14mu}{of}\mspace{14mu}{subband}\mspace{14mu} k}\end{matrix} & (20)\end{matrix}$Received estimates of the transmit data symbols x(s=n₀+1|n, k=n₁+1) ineach layer n are MS decoded in equation (21) to generate the estimatesx(u|n) of the data symbols for u in layer n and prior to MS encoding inthe transmitter. We find

$\begin{matrix}\begin{matrix}{{x\left( u \middle| n \right)} = {N_{c}^{- 1}{\sum\limits_{{n\; 0},{n\; 1}}{{x\left( {{s = \left. {n_{0} + 1} \middle| n \right.},{k = {n_{1} + 1}}} \right)}C*\left( {u_{0} +} \right.}}}} \\\left. {{u_{1}N_{0}},{n_{0} + {n_{1}N_{0}}}} \right) \\{= {{{MS}\mspace{14mu}{decoding}\mspace{14mu}{of}\mspace{14mu}{received}\mspace{14mu}{data}\mspace{14mu}{symbol}\mspace{14mu} s} = {u_{0} + 1}}} \\{{{in}\mspace{14mu}{subband}\mspace{14mu} k} = {u_{1} + 1}} \\{= {{receive}\mspace{14mu}{MS}\mspace{14mu}{decoded}\mspace{14mu}{data}\mspace{14mu}{symbol}\mspace{14mu}{for}\mspace{14mu} u\mspace{14mu}{for}\mspace{14mu}{layer}\mspace{14mu} n}}\end{matrix} & (21)\end{matrix}$

using the orthgonality property of CN _(c) ⁻¹Σ_(n0,n1) C(u ₀ +u ₁ N ₀ ,n ₀ +n ₁ N ₀)C*(u ₀ +u ₁ N ₀ ,n ₀ +n₁ N ₀)=δ(u,u)

-   -   wherein

$\begin{matrix}{C*={{complex}\mspace{14mu}{conjugate}\mspace{14mu}{transpose}\mspace{14mu}{of}\mspace{11mu} C}} \\{{{\delta\left( {\underset{\_}{u},u} \right)} = {{0\mspace{14mu}{for}\mspace{14mu}\underset{\_}{u}} \neq u}},{{delta}\mspace{14mu}{function}}} \\{= {{1\mspace{14mu}{for}\mspace{14mu}\underset{\_}{u}} = u}} \\{\underset{\_}{u} = {{\underset{\_}{u}}_{0} + {{\underset{\_}{u}}_{1}N_{0}}}}\end{matrix}$

Transmitted data symbols x(u|n) for the MS OFDM WiFi QLM mode are thetransmitted data symbols x(s|k) for the OFDM WiFi QLM mode with theequivalence that user indices u₀=0, 1, . . . , N₀−1 and s=1, 2, . . . ,N₀ refer to the same data symbols, and k=1, 2 . . . , N₁ refer to thesame data symbols, and the MS encoding mode is implemented before thetransmit signal processing in FIG. 14 and in equations (11)-(13) and theMS decoding mode is implemented after the receive signal processing inFIG. 15) and in equations (15)-(17) and which means the MS mode does notimpact this transmit and receive signal processing for OFDM WiFi QLM.Fast MS encoding and MS decoding algorithms are available and have thecomputational complexity

$\begin{matrix}\begin{matrix}{{Real}\mspace{14mu}{multiplies}\mspace{14mu} R_{M}} & {R_{M} = {n_{p}\; 2M_{c}N_{c}}} \\{{Real}{\;\mspace{11mu}}{adds}\mspace{14mu} R_{A}} & {R_{A} = {n_{p}3M_{c}N_{c}}}\end{matrix} & (21)\end{matrix}$wherein M_(c)=log₂(N_(c)) and N_(c)=n_(s)−1 for n>1 and for n=1 assumingN_(c)=n_(s)−1 which corresponds to neglecting the last symbol order tohave a fast algorithm for all n.

FIG. 19 is a WiFi transmitter block diagram modified to support a MSOFDM WiFi QLM mode using ¾-data symbol groups to increase the symboltransmission rate and with an increase in transmitter power to supportthis increased data rate. The WiFi standard power spectrum in FIG. 1 ismodified by the OFDM WiFi QLM mode signal processing defined in FIGS.7,14,15 when the ML demodulation is implemented. Signal processingstarts with the stream of user input data words d_(k) 46 with k indexedover the words. Frame processor 47 accepts these data words andtypically performs turbo error correction encoding, error detectioncyclic redundant coding CRC, frame formatting, and passes the outputs tothe symbol encoder 48 which encodes the frame data words into datasymbols and prior to handover to the signal processing 50 the datasymbols are encoded with multi-scale encoding MS in FIG. 16 usinggeneralized complex Walsh codes in FIG. 17 or other orthogonal andquasi-orthogonal codes.

Signal processing 50 in FIG. 19 implements the ML OFDM WiFi QLM modesignal generation in FIG. 7, 14-15 for the n_(p) QLM layers. The QLM MLsignals for each layer is offset in time in each subband filter in 50,are summed in 51 and waveform encoded in 51, the output stream ofcomplex baseband signal samples 52 z(i) defined in 14 in FIG. 15 ishanded over to the digital-to-analog converter DAC which generates theanalog equivalent z(t) of the z(i), and the DAC output analog signalz(t) is single sideband SSB up-converted 52 to RF and transmitted as theanalog signal v(t) wherein v(t) is the real part of the complex basebandsignal z(t) at the RF frequency.

FIG. 19 applies to the OFDM WiFi QLM mode in the absence of MS encodingby deleting the MS encoding in 48.

FIG. 20 is a WiFi receiver block diagram modified to support a MS OFDMWiFi QLM communications signal from the MS OFDM WiFi QLM transmitter inFIG. 19. Receive signal processing for QLM demodulation starts with thewavefront 54 incident at the receiver antenna which forms the receive Rxsignal {circumflex over (ν)}(t) at the antenna output 55 where{circumflex over (ν)}/(t) is an estimate of the transmitted signal v(t)52 in FIG. 19 that is received with errors in time Δt, frequency Δt andphase Δθ and additive noise. This received signal {circumflex over(ν)}(t) is amplified and down-converted to baseband by the analog frontend 56, synchronized (synch.) in time t and frequency f, waveformremoved to detect the received QLM signal at the QLM symbol rate,inphase and quadrature I/Q detected, and analog-to-digital ADC converted57. ADC output signal z(i) is de-multiplexed into n_(p) parallel signals58 which are offset in time by 0, Δt, 2Δt, . . . ,(n_(p)−1) Δt forlayers n=1, 2, . . . , n_(p) wherein Δt=T_(s)/n_(p) and T_(s) is thesymbol interval for each layer, and processed by the pre-summed FFTsubband filter bank 58 to recover the detected correlated data symbolsin each layer. Outputs are ML demodulated 59 to recover estimates of theMS encoded data symbols for each layer for each set of transmitted datasymbols by implementing the ML algorithm 4 in equations (7) for the3-filter and 4-filter groups in each layer. The MS code is removed 59 torecover the data symbols and the outputs are further processed 60,61 torecover estimates {circumflex over (d)}_(k) of the transmitted datawords d_(k) in 46 in FIG. 19.

FIG. 20 applies to the OFDM WiFi QLM mode in the absence of MS decodingby deleting the MS decoding in 59.

This OFDM WIFi QLM architecture applies with parameter changes to theother WiFi options, to WiMax which has a larger frequency band, to theLTE downlink which also uses OFDM, and to the LTE uplink since the QLMarchitecture generates shaped contiguous subbands which are SC-OFDM.

LTE uplink uses SC-OFDM filter banks which are weighted FFT⁻¹ tonefilters with the LTE transmission partitioned into sub-frame and framelengths. LTE filters can be combined into user subbands over thefrequency band with each user subband consisting of the weighted tonefilters over the user subband frequency band. OFDM QLM ML generates thesame weighted FFT⁻¹ tone filters by combining each set of 4 FFT-¹ tonesfor the 64 point FFT⁻¹ for WiFi standard into a weighted subband filterwhich subband filter is a weighted FFT⁻¹ tone filter for a 64/4=16 pointFFT⁻¹. This means the OFDM QLM ML architecture developed in thisspecification directly applies to the LTE uplink with parameter changesfor the weighted FFT⁻¹ filter spacing, number of filters and frequencyband, sub-frame and frame lengths, and communication protocols. Inparticular this means the QLM ML 2,3,4-data group architecture fortransmission of n_(p) layers or channels of QLM communications can beimplemented for the LTE uplink communications and with comparableperformance as the OFDM QLM ML communications assuming the frameefficiency for OFDM QLM ML is the same as implemented on the LTE uplink.LTE downlink uses OFDM and which means the OFDM QLM ML architecture inthis specification is directly applicable to the LTE downlink.

The ML modulation and demodulation architectures and algorithms andimplementations and filings disclosed in this patent for OFDM QLM areexamples of available ML, MAP, trellis data symbol, trellis data bit,recursive relaxation, and other optimization architectures andalgorithms to recover estimates of data symbols from layeredcommunications channels for the plurality of applications withdifferentiating parameters that enable demodulation algorithms torecover estimates of the data symbols and for the trellis algorithms torecover the data in the modulated data symbols, in each of thecommunications layers or channels. This patent covers the plurality ofall of these architectures, algorithms, implementations, and filings forgeneration and for recovery of the data symbols in each of thecommunications layers as well as decoding of the data symbols.

This patent covers the plurality of everything related to QLM generationfor WiFi, WiMax, LTE and OFDM/OFDMA, SC-OFDM waveforms, QLM demodulationfor WiFi, WiMax, LTE and OFDM/OFDM, SC-OFDM waveforms, and data recoveryof QLM and to the corresponding bounds on QLM to all QLM inclusive oftheory, teaching, examples, practice, and of implementations for relatedtechnologies. The representative trellis and ML algorithms for QLMdemodulation are examples to illustrate the methodology and validate theperformance and are representative of all QLM demodulation algorithmsincluding all maximum likelihood ML architectures, maximum a posterioriMAP, maximum a priori, finite field techniques, direct and iterativeestimation techniques, trellis symbol and iterative trellis symbol andwith/without simplifications, trellis bit and iterative trellis bit andwith/without simplifications- and with/without bit error correctioncoding, and all other related algorithms whose principal function is torecover estimates of the transmitted symbols for QLM parallel layeredmodulation as well as data recovery related to QLM and the QLM bounds.

Preferred embodiments in the previous description of modulation anddemodulation algorithms and implementations for QLM for the knownmodulations: and demodulations and for all future modulations anddemodulations, are provided to enable any person skilled in the art tomake or use the present invention. The various modifications to theseembodiments will be readily apparent to those skilled in the art and thegeneric principles defined herein may be applied to other embodimentswithout the use of the inventive faculty. Thus, the present invention isnot intended to be limited to the embodiments shown herein but is to beaccorded the wider scope consistent with the principles and novelfeatures disclosed herein. Additional filings for QLM signal processingand bound include the plurality of information theoretic filings withexamples being radar, imaging, and media processing.

1. A method for implementation of Quadrature Parallel Layered Modulation(QLM) maximum likelihood (ML) communications over orthogonal frequencymultiplexing (OFDM) for WiFi or WiMax, said method comprising the steps:implementing QLM for OFDM transmission over the WiFi or WiMax frequencybands and data packet length by generating a communications signal overa frequency bandwidth at the data symbol rate n_(p)/T_(s) and over thedata packet length with properties 1) each data symbol is encoded withinformation and has the same waveform. 2) the Nyquist rate for the datasymbol transmission is equal to 1/T_(s), 3) the Nyquist rate is equal tothe bandwidth 1/T_(s) of the data symbol waveform and is the data symboltransmission rate 1/T₉ which is sufficient to transmit all of theinformation in each data symbol, 4) n_(p) is the increase in the Nyquistdata rate supported by QLM, 5) this increase in Nyquist data rate can beviewed as increasing to n_(p) the number of parallel communicationschannels supported by QLM at the data symbol rate 1/T_(s), 6) timingoffset ΔT_(s) equal to the data symbol spacing ΔT_(s)=T_(s)/n_(p) is thedifferentiating parameter when viewing this increase in data symbol rateas parallel communications channels which are independent since the QLMdemodulation algorithm recovers the data symbols and data symbol encodedinformation at the QLM data symbol rate n_(p)/T_(s), 7) in the datapacket each of these parallel channels of communications has a uniquetiming offset, channel 1 starts with the first data symbol with nooffset, channel 2 starts with the second data symbol offset by ΔT_(s),channel 3 starts the third data symbol offset by 2ΔT_(s), and continuingto the channel n, data symbol with (n_(p)−1) ΔT_(s) offset, 8) in thedata packet each of these parallel channels of communications continuesuntil the end of the packet which means channel 1 continues to the lastdata symbol in the data packet, and channels 2 thru (n_(p)−1)continue tothe second last data symbol in the data packet, and 9) which implementsthe QLM signal comprising n_(p) parallel layers of communicationschannels over the same data symbol frequency bandwidth and over the samedata packet length, starting the QLM transmission using a set ofcontiguous subbands over the OFDM WiFi or WiMax frequency band B anddata packet length with each subband occupying the frequency bandassigned to a subset of OFDM tones, using a subset of these subbands asdata subbands for transmission of the QLM data symbols over the WiFi orWiMax data packet length, generating a first communications signal overthe first layer or channel consisting of the data symbols over the ML2,3,4-data symbol groups in each data subband and disclosed in thespecification, at the carrier frequency for the subband by modulating afirst set of data symbols with a waveform at a 1/T_(s) symbol ratewherein “T_(s)” is the time interval between contiguous symbols and1/T_(s) is equal to the Nyquist rate 1/T_(s)=ΔB for subband spacing ΔB,generating a second communications signal over the second layer orchannel consisting of the data symbols over the ML 2,3,4-data symbolgroups in each data subband at the same carrier frequency for eachsubband by modulating a second set of data symbols with the samewaveform at the same symbol rate as the first stream of data symbols andwith a time offset ΔT_(s) equal to ΔT_(s)=T_(s)/n_(p) wherein “n_(p)” isthe number of QLM layers or channels in each subband in the OFDM WiFi orWiMax data packet, for any additional layers or channels, continuinggeneration of communication signals over the additional layers orchannels consisting of the data symbols over the ML 2,3,4-data symbolgroups in each data subband at the same carrier frequency for eachsubband by modulating additional sets of data symbols with the samewaveform at the same data symbol rate as the first and second streams ofdata symbols, with time offsets increasing in each communication signalin increments of ΔT_(s)=T_(s)/n_(p) until the n_(p) communicationssignals are generated for n, QLM layers or channels, generating said QLMML communications signals consisting of the n_(p) layere or channels ofdata symbols over the ML 2,3,4-data symbol groups in each data subbandby implementing a computationally efficient post-weighted inverse fastFourier transform (FFT⁻¹)algorithm, transmitting said QLM MLcommunications signal consisting of the n_(p) QLM layers or channels ina QLM communications transmitter, receiving said QLM ML communicationssignals in a receiver and processing said communications signals togenerate a set of contiguous subbands over the OFDM WiFi or WiMaxfrequency band B and data packet length with each subband occupying thefrequency band assigned to a subset of OFDM tones, detecting a first setof received correlated data symbols for the first communications layeror channel over the ML 2,3,4-data symbol groups in each data subband ineach data packet length by convolving the received communications signalin the data subband with the complex conjugate of the transmitted datasymbol subband waveform at the data symbol spacing T_(s) andsynchronized in time to the received set of data symbols in the first ofthe transmitted communications layers or channels and wherein thecorrelation in the received set of data symbols is caused by the overlapof the transmitted data symbols, detecting a second set of receivedcorrelated data symbols for the second communications layer or channelover the ML 2,3,4-data symbol groups in each data subband in each datapacket length by convolving the received communications signal in thedata subband with the complex conjugate of the transmitted data symbolsubband waveform at the data symbol spacing T_(s) and starting with thetime offset offset ΔT_(s) and synchronized in time with the receivedcorrelated data symbols in the second of the transmitted communicationslayers or channels, for any additional transmitted communications layersor channels, continuing detection of the received sets of correlateddata symbols over the ML 2,3,4-data symbol groups in each data packetlength by convolving the received communications signal in the datasubband with the complex conjugate of the transmitted data symbolsubband waveform at the data symbol spacing T_(s) and starting with thetime offset offseta which are incremented by ΔT_(s)=T_(s)/n_(p) untilthe communications signals are detected for n_(p) QLM layers orchannels, implementing said QLM ML receive n_(p) sets of correlated datasymbol detections with each set corresponding to one of the n_(p)communications layers or channels of data symbols over the ML 2,3,4-datasymbol groups in each data subband by implementing a computationallyefficient pre-weighted fast Fourier transform (FFT)algorithm disclosedin the specification, recovering the transmitted data symbols in thereceiver by demodulating the detected n, sets of correlated data symbolsin a receiver using a ML demodulation algorithm, and combining saidalgorithm with error correction decoding to recover the transmittedinformation in the detected data symbols.
 2. The method of claim 1wherein the QLM communication signals have the following communicationslink performance properties: maximum capacity “C” in bits/second isdefined by equationC=max{n _(p) B log₂(1+(S/N)/n _(p)^2)} wherein the maximum “max” is withrespect to n_(p), “log₂” is the logarithm to the base 2, “B” is thefrequency bandwidth in Hz, and “S/N” is the ratio signal-to-noise over“B”, maximum number of bits “b” per symbol interval T_(s)=1/B is definedby equationmax{b}=max{n _(p)(1+(S/N)/n _(p)^2)}, maximum communications efficiency“Ti” in Bits/second/Hz is defined by equationmax(η)=max{b}, minimum signal-to-noise ratio per bit “E_(b)/N_(o)” isdefined by equationmin{E _(b) /N _(o)}=min{[n _(p)^2/b][2^b}/n _(p)−1]} wherein “E_(b)” isthe energy per bit, “N_(a)” is the power spectral density of the noise,and the minimum “min” is the minimum with respect to n_(p), maximum datasymbol rate “n_(p)/T_(s)” is defined by equation $\begin{matrix}{{\max\left\{ {n_{p}/T_{s}} \right\}} = {n_{p}B}} \\{{= {n_{p}\left( {{Nyquist}\mspace{14mu}{rate}} \right)}},{and}}\end{matrix}$ wherein these performance bounds apply to a communicationsreceiver demodulation performance of a QLM communications linkconsisting of n_(p) QLM layers or channels and enable the design andimplementation of QLM communications over communications links withperformance limited by these bounds.
 3. The method of claim 1 or 2wherein the QLM communications systems design and implementation use theQLM design properties: QLM E_(b)/N_(o) is the ratio of energy perinformation bit E_(b) to the noise power density N_(o) and is scaled bythe number of QLM layers n_(p) to derive the value[E_(b)/N_(o)]=E_(b)/N_(o)/n_(p) required to support the samebit-error-rate BER as a single layer with E_(b)/N_(o)=[E_(b)/N_(o)], QLMS/N is the ratio of signal power S to the noise power N and is scaled bythe square of the number of QLM layers n_(p) to derive the value[S/N]=S/N/n_(p)^2 required to support the same BER as a single layerwith S/N=[S/N], these QLM scaling laws are bounds on the QLM performancein claim 2 and which bounds are approximated by the QLM ML demodulationperformance, these QLM scaling laws when combined with data symbolmodulation and measured BER performance enable estimation of the QLM MLdemodulation performance which is the number of bits “b” per symbolinterval that can be supported for E_(b)/N_(o) and S/N, and whereinthese QLM scaling laws and parameters are incorporated into the designand implementation of QLM communications transmitters and communicationsreceivers for communications links.
 4. The method of claim 1 wherein thetransmitted communications QLM ML data symbols for each layer or channelfor OFDM WiFi or WiMax are encoded with a multi-scale (MS) code toimprove the bit-error-rate (BER) performance, said transmitter encodingand receiver decoding design and implementations comprise the steps:identifying the set of N₁ data subbands and the number N₀ of datasymbols in each data subband for each QLM layer or channel ofcommunications, generating a code division multiple access (CDMA) codewith a code length N_(c) equal to a product of a number of chips N_(o)for a first scale CDMA encoding having first code and chip indices usedto encode data symbols within each data subband and a number of chips N₁for a second scale CDMA encoding having second code and chip indicesused to encode data symbols over the entire set of N₁ data subbands,forming a 2-scale CDMA code by assigning code and chip indices such thatthe 2-scale CDMA code and chip indices are the algebraic addition of thefirst code and chip indices plus scaled second code and chip indiceswherein said scaled second code and chip indices are generated using ascale factor that comprises the number of indices in the first scaleCDMA code, wherein the steps of generating and forming further includeencoding data symbols with the 2-scale CDMA code to generate encodedchips, and assigning each of the encoded chips to a data subband inaccordance with the second CDMA code indices, and assigning each encodedchip to a chip position within its assigned data subband in accordancewith the first CDMA code indices by implementing the MS encoding inequation (20) in the specification to generate the encoded data symbolsfor each QLM layer or channel, using these encoded data symbols togenerate the QLM communications signals for transmission by thetransmitter, by implementing a computationally efficient post-weightedinverse fast Fourier transform (FFT⁻¹)algorithm, transmitting saidcommunications signals consisting of the n_(p) QLM layers or channels ina QLM communications transmitter, receiving said communications signalsin a receiver and processing said communications signals to generate aset of contiguous subbands over the OFDM WiFi or WiMax frequency band Band data packet length with each subband occupying the frequency bandassigned to a subset of OFDM tones, implementing the QLM receive signalprocessing to detect the n_(p) sets of correlated data symbol with eachset corresponding to one of the n_(p) communications layers or channelsby implementing a computationally efficient pre-weighted fast Fouriertransform (FFT)algorithm disclosed in the specification, recovering thetransmitted data symbols in the receiver for by demodulating thedetected n_(p) sets of correlated data symbols in a receiver using ademodulation algorithm, decoding these demodulated data symbols byimplementing the MS decoding in equation (21) in the specification, andcombining said MS decoding with error correction decoding to recover thetransmitted information.
 5. The method of claim 1, wherein the symboldemodulation in the receiver is designed and implemented with a maximumlikelihood (ML) algorithm, comprising the steps of: implementing the QLMML receive signal processing of the transmitted OFDM WiFi or WiMax QLMML communications consisting of the n, layere or channels of datasymbols over the ML 2,3,4-data symbol groups in each data subband todetect the n_(p) layers or channels of correlated data symbols byimplementing a computationally efficient pre-weighted fast Fouriertransform (FFT)algorithm disclosed in the specification, measuring the nby n correlation matrix H whose elements are the correlationcoefficients between the n data symbols in each subband for the n_(p)layers or channels of communications, organizing each set of n detectedcorrelated data symbols for each subband into a n×1 row vector or 1×ncolumn vector which is the detected signal vector Y whose elements arethe n detected correlated data symbols, organizing the set of nestimated transmit data symbols for each subband into a n×1 row vectoror n×1 column vector X which is the ML solution, evaluating the MLsolution X=H⁻¹Y in equation (7) in the specification wherein H⁻¹ is thematrix inverse of H for each subband, and using error correctiondecoding to recover the transmitted information in the encoded datasymbols recovered by the ML algorithm.
 6. The method of claim 1, furthercomprising a method for design and implementation of QLM ML for SingleCarrier OFDM (SC-OFDM) for long-term evolution (LTE) uplinkcommunications and OFDM QLM ML for LTE downlink communications, saidmethod comprising the steps: implementing QLM ML for SC-OFDM uplinktransmission over the LTE frequency band and data sub-frame and framelengths, using the QLM ML subband architecture to generate thesingle-carrier frequency filters which are the subbands in claim 1 andwhich subbands can be combined to form the LTE user subbands thatpartition the frequency band into larger user subbands and bytransmitting over the LTE sub-frames and frames using the QLM ML2,3,4-data symbol groups supporting n, layers or channels ofcommunications, generating the n_(p) layers or channels ofcommunications over each user subband and data symbol group byImplementing a computationally efficient post-weighted inverse fastFourier transform (FFT⁻¹) algorithm, transmitting said communicationssignals in a LTE communications transmitter on the uplink, receivingsaid communications signals in a receiver and processing saidcommunications signals to generate each set of user subbands using thesubband architecture in claim 1 for each data sub-frame and frame anddetecting the transmitted data symbols in each QLM ML 2,3,4-data symbolgroup supporting n_(p) layers or channels of communications byimplementing said QLM receive correlated data symbol detections witheach set corresponding to one of the communications layers or channelsusing a computationally efficient pre-weighted fast Fourier transform(FFT)algorithm, recovering the transmitted data symbols in the receiverby demodulating the detected n_(p) sets of correlated data symbols in areceiver using a QLM ML demodulation algorithm, combining said algorithmwith error correction decoding to recover the transmitted information,implementing OFDM QLM ML downlink communications using the LTE frequencyband and data sub-frames for downlink transmission, implementing saiddownlink transmission in a transmitter, receiving said communicationssignals in a receiver and processing said OFDM QLM ML communications torecover estimates of the transmitted data symbols, and using errorcorrection decoding to recover the transmitted information.